Dana L. Chesney

Evidence for a shared mechanism used in multiple-object tracking and subitizing

It has been proposed that the mechanism that supports the ability to keep track of multiple moving objects also supports subitizing—the ability to quickly and accurately enumerate a small set of objects. To test this hypothesis, we investigated the effects on subitizing when human observers were required to perform a multiple object tracking task and an enumeration task simultaneously. In three experiments, participants (Exp. 1, N = 24; Exp. 2, N = 11; Exp. 3, N = 37) enumerated sets of zero to nine squares that were flashed while they tracked zero, two, or four moving discs. The results indicated that the number of items participants could subitize decreased by one for each item they tracked. No such pattern was seen when the enumeration task was paired with an equally difficult, but nonvisual, working memory task. These results suggest that a shared visual mechanism supports multiple object tracking and subitizing.

Knowledge on the line: Manipulating beliefs about the magnitudes of symbolic numbers affects the linearity of line estimation tasks

It has been suggested that differences in performance on number-line estimation tasks are indicative of fundamental differences in people’s underlying representations of numerical magnitude. However, we were able to induce logarithmic-looking performance in adults for magnitude ranges over which they can typically perform linearly by manipulating their familiarity with the symbolic number formats that we used for the stimuli. This serves as an existence proof that individuals’ performances on number-line estimation tasks do not necessarily reflect the functional form of their underlying numerical magnitude representations. Rather, performance differences may result from symbolic difficulties (i.e., number-to-symbol mappings), independently of the underlying functional form. We demonstrated that number-line estimates that are well fit by logarithmic functions need not be produced by logarithmic functions. These findings led us to question the validity of considering logarithmic-looking performance on number-line estimation tasks as being indicative that magnitudes are being represented logarithmically, particularly when symbolic understanding is in question.

An eye for relations: eye-tracking indicates long-term negative effects of operational thinking on understanding of math equivalence.

Prior knowledge in the domain of mathematics can sometimes interfere with learning and performance in that domain. One of the best examples of this phenomenon is in students’ difficulties solving equations with operations on both sides of the equal sign. Elementary school children in the U.S. typically acquire incorrect, operational schemata rather than correct, relational schemata for interpreting equations. Researchers have argued that these operational schemata are never unlearned and can continue to affect performance for years to come, even after relational schemata are learned. In the present study, we investigated whether and how operational schemata negatively affect undergraduates’ performance on equations. We monitored the eye movements of 64 undergraduate students while they solved a set of equations that are typically used to assess children’s adherence to operational schemata (e.g., 3 + 4 + 5 = 3 + __). Participants did not perform at ceiling on these equations, particularly when under time pressure. Converging evidence from performance and eye movements showed that operational schemata are sometimes activated instead of relational schemata. Eye movement patterns reflective of the activation of relational schemata were specifically lacking when participants solved equations by adding up all the numbers or adding the numbers before the equal sign, but not when they used other types of incorrect strategies. These findings demonstrate that the negative effects of acquiring operational schemata extend far beyond elementary school.

Visual nesting impacts approximate number system estimation.

The approximate number system (ANS) allows people to quickly but inaccurately enumerate large sets without counting. One popular account of the ANS is known as the accumulator model. This model posits that the ANS acts analogously to a graduated cylinder to which one “cup” is added for each item in the set, with set numerosity read from the “height” of the cylinder. Under this model, one would predict that if all the to-be-enumerated items were not collected into the accumulator, either the sets would be underestimated, or the misses would need to be corrected by a subsequent process, leading to longer reaction times. In this experiment, we tested whether such miss effects occur. Fifty participants judged numerosities of briefly presented sets of circles. In some conditions, circles were arranged such that some were inside others. This circle nesting was expected to increase the miss rate, since previous research had indicated that items in nested configurations cannot be preattentively individuated in parallel. Logically, items in a set that cannot be simultaneously individuated cannot be simultaneously added to an accumulator. Participants’ response times were longer and their estimations were lower for sets whose configurations yielded greater levels of nesting. The level of nesting in a display influenced estimation independently of the total number of items present. This indicates that miss effects, predicted by the accumulator model, are indeed seen in ANS estimation. We speculate that ANS biases might, in turn, influence cognition and behavior, perhaps by influencing which kinds of sets are spontaneously counted.

Task Constraints Affect Mapping From Approximate Number System Estimates to Symbolic Numbers

The Approximate Number System (ANS) allows individuals to assess nonsymbolic numerical magnitudes (e.g., the number of apples on a tree) without counting. Several prominent theories posit that human understanding of symbolic numbers is based – at least in part – on mapping number symbols (e.g., 14) to their ANS-processed nonsymbolic analogs. Number-line estimation – where participants place numerical values on a bounded number-line – has become a key task used in research on this mapping. However, some research suggests that such number-line estimation tasks are actually proportion judgment tasks, as number-line estimation requires people to estimate the magnitude of the to-be-placed value, relative to set upper and lower endpoints, and thus do not so directly reflect magnitude representations. Here, we extend this work, assessing performance on nonsymbolic tasks that should more directly interface with the ANS. We compared adults’ (n = 31) performance when placing nonsymbolic numerosities (dot arrays) on number-lines to their performance with the same stimuli on two other tasks: Free estimation tasks where participants simply estimate the cardinality of dot arrays, and ratio estimation tasks where participants estimate the ratio instantiated by a pair of arrays. We found that performance on these tasks was quite different, with number-line and ratio estimation tasks failing to the show classic psychophysical error patterns of scalar variability seen in the free estimation task. We conclude the constraints of tasks using stimuli that access the ANS lead to considerably different mapping performance and that these differences must be accounted for when evaluating theories of numerical cognition. Additionally, participants showed typical underestimation patterns in the free estimation task, but were quite accurate on the ratio task. We discuss potential implications of these findings for theories regarding the mapping between ANS magnitudes and symbolic numbers.