Is number visual? Is vision numerical? Investigating the relationship between visual representations and the property of magnitude
Friday, May 8, 1:00 – 3:00 pm
Royal Ballroom 6-8
Organizer: Michael C. Frank (Massachusetts Institute of Technology)
Presenters: David Burr (Dipartimento di Psicologia, Università Degli Studi di Firenze and Department of Psychology, University of Western Australia), Michael C. Frank (Massachusetts Institute of Technology), Franconeri, Steven (Northwestern University), David Barner (University of California, San Diego), Justin Halberda (Johns Hopkins University)
The ability to manipulate exact numbers is a signature human achievement, supporting activities like building bridges, designing computers, and conducting economic transactions. Underlying this ability and supporting its acquisition is an evolutionarily-conserved mechanism for the manipulation of approximate quantity: the analog magnitude system. The behavioral and neural signatures of magnitude representations have been extensively characterized but how these representations interact with other aspects of cognitive and visual processing is still largely unknown. Do magnitude features attach to objects, scenes, or surfaces? Is approximate magnitude representation maintained even for sets for which exact quantity is known? Is magnitude estimation ability altered by experience?
The goal of our symposium is to look for answers to these questions by asking both how number is integrated into visual processing and how visual processing in turn forms a basis for the acquisition and processing of exact number. We address these questions through talks on three issues: 1) the basic psychophysical properties of numerical representations (Halberda, Burr), 2) how visual mechanisms integrate representations of number (Franconeri & Alvarez), and 3) how these representations support exact computation, both in standard linguistic representations (Frank) and via alternative representations (Barner).
The issues addressed by our symposium have been a focus of intense recent interest. Within the last four years there have been a wide variety of high-profile reports from developmental, neuroscientific, comparative, and cross-linguistic/cross-cultural studies of number. Research on number is one of the fastest moving fields in cognitive science, due both to the well-defined questions that motivate research in this field and to the wide variety of methods that can be brought to bear on these questions.
The target audience of our symposium is a broad group of vision scientists, both students and faculty, who are interested in connecting serious vision science with cognitive issues of broad relevance to a wide range of communities in psychology, neuroscience, and education. In addition, the study of number provides an opportunity to link innovations in vision research methods—including psychophysical-style experimental designs, precise neuroimaging methods, and detailed computational data analysis—with deep cognitive questions about the nature of human knowledge. We anticipate that attendees of our symposium will come away with a good grasp of the current state of the art and the outstanding issues in the interface of visual and numerical processing.
A visual sense of number
Evidence exists for a non-verbal capacity to apprehend number, in humans (including infants), and in other primates. We investigated numerosity perception in adult humans, by measuring Weber fractions with a series of techniques, and by adaptation. The Weber fraction measurements suggest that number estimation and “subitizing” share common mechanisms. Adapting to large numbers of dots increased apparent numerosity (by a factor of 2-3), and adapting to small numbers increased it. The magnitude of adaptation depended primarily on the numerosity of the adapter, not on size, orientation or contrast of test or adapter, and occurred with very low adapter contrasts. Varying pixel density had no effect on adaptation, showing that it depended solely on numerosity, not related visual properties like texture density. We propose that just as we have a direct visual sense of the reddishness of half a dozen ripe cherries so we do of their sixishness. In other words there are distinct qualia for numerosity, as there are for colour, brightness and contrast, not reducible to spatial frequency or density of texture.
Language as a link between exact number and approximate magnitude
Michael C. Frank
Is exact number a human universal? Cross-cultural fieldwork has given strong evidence that language for exact number is an invention which is not present in all societies. This result suggests a range of questions about how learning an exact number system may interact with pre-existing analog magnitude representations. More generally, number presents a tractable case of the Whorfian question of whether speakers of different languages differ in their cognition. We addressed these questions by studying the performance of the Pirahã, an Amazonian group in Brazil, on a range of simple quantity matching tasks (first used by Gordon, 2004). We compared the performance of this group to the performance of English-speakers who were unable to use exact numerical representations due to a concurrent verbal interference task. We found that both groups were able to complete simple one-to-one matching tasks even without words for numbers and both groups relied on analog magnitude representations when faced with a more difficult task in which items in the set to be estimated were presented one at a time. However, performance between the two groups diverged on tasks in which other strategies could be used. We conclude that language for number is a “cognitive technology” which allows the manipulation of exact quantities across time, space, and changes in modality, but does not eliminate or substantially alter users’ underlying numerical abilities.
Rapid enumeration is based on a segmented visual scene
Steve Franconeri, George Alvarez
How do we estimate the number of objects in a set? One primary question is whether our estimates are based on an unbroken visual image or a segmented collection of discrete objects. We manipulated whether individual objects were isolated from each other, or grouped into pairs by irrelevant lines. If number estimation operates over an unbroken image, then this manipulation should not affect estimates. But if number estimation relies on a segmented image, then grouping pairs of objects into single units should lead to lower estimates. In Experiment 1, participants underestimated the number of grouped squares, relative to when the connecting lines were ‘broken’. Experiment 2 presents evidence that this segmentation process occurred broadly across the entire set of objects. In Experiment 3, a staircase procedure provides a quantitative measure of the underestimation effect. Experiment 4 shows that is the strength of the grouping effect was equally strong for a single thin line, and the effect can be eliminated by a tiny break in the line. These results provide the first direct evidence that number estimation relies on a segmented input.
Constructing exact number approximately: a case study of mental abacus representations
Exact numerical representation is usually accomplished through linguistic representations. However, an alternative route for accomplishing this task is through the use of a “mental abacus”—a mental image of an abacus (a device used in some cultures for keeping track of exact quantities and doing arithmetic via the positions of beads on a rigid frame). We investigated the nature of mental abacus representations by studying children ages 7-15 who were trained in this technique. We compared their ability to read the cardinality of “abacus flashcards” (briefly presented images of abacuses in different configurations) with their ability to enumerate sets of dots after similarly brief, masked presentation. We conducted five studies comparing abacus flashcards to: (1) random dot enumeration, (2) spatially proximate dot enumeration, (3) enumeration of dots arranged in an abacus configuration without the abacus frame, (4) enumeration of dots on a rotated abacus, (5) enumeration of dots arranged on an abacus. In all conditions, participants were faster and more accurate in identifying the cardinality of an abacus than they were in enumerating the same number of beads, even when the display was physically identical. Analysis of errors suggested that children in our studies viewed the abacus as a set of objects with each separate row of beads being a single object, each with its own independent magnitude feature. Thus, the “mental abacus” draws on pre-existing approximate and exact visual abilities to construct a highly accurate system for representing large exact number.
An interface between vision and numerical cognition
While the similarity of numerical processing across different modalities (e.g., visual objects, auditory objects, extended visual events) suggests that number concepts are domain general even at the earliest ages (4 month old babies), visual processing is constrained in ways that may have constrained the numerical concepts humans have developed. In this talk I discuss how online processing of numerical content is shaped by the constraints of both object-based and ensemble-based visual processing and discuss how numerical content and vision engage one another.