Quantity Judgments Research Articles

Transfer of cognitive training across magnitude dimensions achieved with concurrent brain stimulation of the parietal lobe.

Improvement in performance following cognitive training is known to be further enhanced when coupled with brain stimulation. Here we ask whether training-induced changes can be maintained long term and, crucially, whether they can extend to other related but untrained skills. We trained overall 40 human participants on a simple and well established paradigm assessing the ability to discriminate numerosity--or the number of items in a set--which is thought to rely on an "approximate number sense" (ANS) associated with parietal lobes. We coupled training with parietal stimulation in the form of transcranial random noise stimulation (tRNS), a noninvasive technique that modulates neural activity. This yielded significantly better and longer lasting improvement (up to 16 weeks post-training) of the precision of the ANS compared with cognitive training in absence of stimulation, stimulation in absence of cognitive training, and cognitive training coupled to stimulation to a control site (motor areas). Critically, only ANS improvement induced by parietal tRNS + Training transferred to proficiency in other parietal lobe-based quantity judgment, i.e., time and space discrimination, but not to quantity-unrelated tasks measuring attention, executive functions, and visual pattern recognition. These results indicate that coupling intensive cognitive training with tRNS to critical brain regions resulted not only in the greatest and longer lasting improvement of numerosity discrimination, but importantly in this enhancement being transferable when trained and untrained abilities are carefully chosen to share common cognitive and neuronal components.

What counts for ‘counting’? Chimpanzees, Pan troglodytes, respond appropriately to relevant and irrelevant information in a quantity judgment task

Nonhuman animals quantify all manner of things, and the way in which this is done is fairly well understood. However, little research has been conducted to determine how they know what is or is not relevant in the instances in which they quantify stimuli. We assessed how four chimpanzees chose between two sets of food items when the items were distributed across separate spatial arrays. Each item was covered by a container, and then was revealed in sequence so that neither whole set was visible at one time. After all containers were revealed, some were revealed again. The chimpanzees should have ignored items that were seen a second time and instead enumerated each item only once. In another test, some of the items were transposed in location and then uncovered again. Here, the chimpanzees needed to recognize that the newly shown food items were ones they already had seen. Overall, the chimpanzees were successful in selecting the truly larger array of items despite these potential distracting re-presentations of items. Discrimination performance also reflected analogue magnitude estimation because comparisons of sets that differed by larger amounts were easier than comparisons that differed by smaller amounts. Thus, chimpanzee quantity judgments for nonvisible sets of items are inexact, but they include an aspect of control for determining when items are uniquely presented versus re-presented.

Relative quantity judgments in the beluga whale (Delphinapterus leucas) and the bottlenose dolphin (Tursiops truncatus)

Numerous studies have documented the ability of many species to make relative quantity judgments using an analogue magnitude system. We investigated whether one beluga whale, Delphinapterus leucas, and three bottlenose dolphins, Tursiops truncatus, were capable of selecting the larger of two sets of quantities, and analyzed if their performance matched predictions from the object file model versus the analog accumulator model. In Experiment 1, the two sets were presented simultaneously, under water, and they were visually (condition 1) or echoically (condition 2) available at the time of choice. In experiment 2, the two sets were presented above the water, successively (condition 1) or sequentially, item-by-item (condition 2), so that they were not visually available at the time of choice (condition 1) or at any time throughout the experiment (condition 2). We analyzed the effect of the ratio between quantities, the difference between quantities, and the total number of items presented on the subjects’ choices. All subjects selected the larger of the two sets of quantities above chance levels in all conditions. However, unlike most previous studies, the subjects’ choices did not match the predictions from the accumulator model. Whether these findings reflect interspecies differences in the mechanisms which underpin relative quantity judgments remains to be determined.

Quantity judgments in the context of risk/reward decision making in striped field mice: first “count,” then hunt

We simulated the situation of risky hunting in the striped field mouse Apodemus agrarius in order to examine whether these animals are able to make a choice between small and large quantities of live prey (ants). In the first (preliminary) experiment we investigated to what extent mice were interested in ants as a live prey and how their hunting activity depended on the quantity of these edible but rather aggressive insects. We placed mice one by one into arenas together with ant groups of different quantities, from 10 to 60. Surprisingly, animals, both wild-caught and laboratory-reared, displayed rather skilled predatory attacks: mice killed and ate from 0.37 ± 003 to 4 ± 0.5 ants per minute. However, there was a threshold number of ants in the arenas when rodents expressed signs of discomfort and started to panic, likely because ants bit them. This threshold corresponds to the dynamic density (about 400 individuals per m2 per min) in the vicinity of anthills and ants' routes in natural environment. In the second experiment mice had to choose between different quantities of ants placed in two transparent tunnels. Ants here served both as food items and as a source of danger. As far as we know, this is the first experimental paradigm based on evaluation of quantity judgments in the context of risk/reward decision making where the animals face a trade-off between the hedonistic value of the prey and the danger it presents. We found that when mice have to choose between 5 vs. 15, 5 vs. 30, and 10 vs. 30 ants, they always tend to prefer the smaller quantity, thus displaying the capacity for distinguishing more from less in order to ensure comfortable hunting. The results of this study are ecologically relevant as they reflect situations and challenges faced by free-living small rodents.

Number without language: comparative psychology and the evolution of numerical cognition

Number without language: comparative psychology and the evolution of numerical cognition

A Pilot Study: Designing and Testing the Task Parameters

The aim of this pilot study was to test the validity, efficiency and measurements of a two-digit number comparison study. Number perception, its neural basis and its relationship to how numerical stimuli are presented have been challenging research topics in cognitive neuroscience for many years. A primary question that has been addressed is whether the perception of the quantity of a visually presented number stimulus is dissociable from its early visual perception. The present study examined the possible influence of visual quality judgment on quantity judgments of numbers. To address this issue, a two-digit number comparison task was developed. We suggest that the design parameters affect the accuracy of the results.

Large quantity discrimination by North Island robins (Petroica longipes)

While numerosity-representation and enumeration of different numbers of objects-and quantity discrimination in particular have been studied in a wide range of species, very little is known about the numerical abilities of animals in the wild. This study examined spontaneous relative quantity judgments (RQJs) by wild North Island robins (Petroica longipes) of New Zealand. In Experiment 1, robins were tested on a range of numerical values of up to 14 versus 16 items, which were sequentially presented and hidden. In Experiment 2, the same numerical contrasts were tested on a different group of subjects but quantities were presented as whole visible sets. Experiment 3 involved whole visible sets that comprised of exceedingly large quantities of up to 56 versus 64 items. While robins shared with other species a ratio-based representation system for representing very large values, they also appeared to have developed an object indexing system with an extended upper limit (well beyond 4) that may be an evolutionary response to ecological challenges faced by scatter-hoarding birds. These results suggest that cognitive mechanism influencing an understanding of physical quantity may be deployed more flexibly in some contexts than previously thought, and are discussed in light of findings across other mammalian and avian species.

Putting the elephant back in the herd: elephant relative quantity judgments match those of other species

The ability to discriminate between quantities has been observed in many species. Typically, when an animal is given a choice between two sets of food, accurate performance (i.e., choosing the larger amount) decreases as the ratio between two quantities increases. A recent study reported that elephants did not exhibit ratio effects, suggesting that elephants may process quantitative information in a qualitatively different way from all other nonhuman species that have been tested (Irie-Sugimoto et al. in Anim Cogn 12:193-199, 2009). However, the results of this study were confounded by several methodological issues. We tested two African elephants (Loxodonta africana) to more thoroughly investigate relative quantity judgment in this species. In contrast to the previous study, we found evidence of ratio effects for visible and nonvisible sequentially presented sets of food. Thus, elephants appear to represent and compare quantities in much the same way as other species, including humans when they are prevented from counting. Performance supports an accumulator model in which quantities are represented as analog magnitudes. Furthermore, we found no effect of absolute magnitude on performance, providing support against an object-file model explanation of quantity judgment.

Simultaneous and sequential comparisons of food quantity and consumption

Two experiments explored how food-quantity comparisons affect judgments of food quantity and consumption. In Experiment 1, participants viewed drawings of portions of mashed potatoes presented either simultaneously or sequentially and later recalled sizes from memory. The observed biases in memory for size were consistent with previous research on the biases in judgments of the sizes of geometric shapes when shapes were presented simultaneously or sequentially. In Experiment 2, participants viewed two serving bowls of pasta salad presented either simultaneously or sequentially and consumed as much or as little as they wished from the smaller bowl. Consistent with previous findings suggesting that the quantities people consume are inversely related to biases in judgments of quantity, the effects observed in Experiment 1 were flipped in Experiment 2. These results suggest that food consumption is likely to be affected by comparison quantities and the effects of comparison quantities will depend on whether quantities are presented simultaneously or sequentially.

Humans and monkeys show similar skill in estimating uncertain outcomes

When information is incomplete but a choice must be made, individuals sometimes can rely on past experiences to help them assess uncertain outcomes in terms of the probabilities of payoffs. Monkeys (Cebus apella) and humans (Homo sapiens) were presented with a test in which they first made quantity judgments between two clear options. Then they made choices in which only one option was visible, and they had to estimate the quantity for the other option. Both species were guided by the past outcomes, as they shifted from selecting the known option to selecting the unknown option at the point at which the known option went from being greater than the average rate of return to being less than the average rate of return. This comparability across species suggests that tallying ongoing average rates of return from repeated choices occurs spontaneously and likely serves an adaptive purpose when dealing with uncertainty in the environment.

Do Social Conditions Affect Capuchin Monkeys’ (Cebus apella) Choices in a Quantity Judgment Task?

Beran et al. (2012) reported that capuchin monkeys closely matched the performance of humans in a quantity judgment test in which information was incomplete but a judgment still had to be made. In each test session, subjects first made quantity judgments between two known options. Then, they made choices where only one option was visible. Both humans and capuchin monkeys were guided by past outcomes, as they shifted from selecting a known option to selecting an unknown option at the point at which the known option went from being more than the average rate of return to less than the average rate of return from earlier choices in the test session. Here, we expanded this assessment of what guides quantity judgment choice behavior in the face of incomplete information to include manipulations to the unselected quantity. We manipulated the unchosen set in two ways: first, we showed the monkeys what they did not get (the unchosen set), anticipating that “losses” would weigh heavily on subsequent trials in which the same known quantity was presented. Second, we sometimes gave the unchosen set to another monkey, anticipating that this social manipulation might influence the risk-taking responses of the focal monkey when faced with incomplete information. However, neither manipulation caused difficulty for the monkeys who instead continued to use the rational strategy of choosing known sets when they were as large as or larger than the average rate of return in the session, and choosing the unknown (riskier) set when the known set was not sufficiently large. As in past experiments, this was true across a variety of daily ranges of quantities, indicating that monkeys were not using some absolute quantity as a threshold for selecting (or not) the known set, but instead continued to use the daily average rate of return to determine when to choose the known versus the unknown quantity.

Quantity judgments of auditory and visual stimuli by chimpanzees (Pan troglodytes).

Many species can choose between two visual sets of stimuli on the basis of quantity. This is true when sets are both visible, or are presented one set at a time or even one item at a time. However, we know comparatively little about how well nonhuman animals can compare auditory quantities. Here, three chimpanzees (Pan troglodytes) chose between two sets of food items when they only heard each item fall into different containers rather than seeing those items. This method prevented the chimpanzees from summing the amount of visible food they saw because there were no visual cues. Chimpanzees performed well, and their performance matched that of previous experiments with regard to obeying Weber's law. They also performed well with comparisons between a sequentially presented auditory set and a fully visible set, demonstrating that duration of presentation was not being used as a cue. In addition, they accommodated empty sets into these judgments, although not perfectly. Thus, chimpanzees can judge auditory quantities in flexible ways that show many similarities to how they compare visual quantities.

The Importance of Replication in Comparative Psychology: The Lesson of Elephant Quantity Judgments

The Importance of Replication in Comparative Psychology: The Lesson of Elephant Quantity Judgments

Comparing children’s Homo sapiens and chimpanzees’ Pan troglodytes quantity judgments of sequentially presented sets of items

Abstract We presented a quantity judgment task that involved comparing two sequentially presented sets of items to preschoolers and chimpanzees using nearly identical procedures that excluded verbal instructions to children. Trial difficulty in this task reflected the ratio difference between sets of discrete items where larger ratios (e.g., 0.80 as from comparing 4 to 5) were more difficult than smaller ones (e.g., 0.50 as from comparing 4 to 8). Children also completed verbal-based tasks probing the relationship between counting proficiency and performance on the quantity judgment task of sequentially presented identical sized items. Both species’ performance was best when ratios between comparison sets were small regardless of set size in all types of tasks. Generally, chimpanzees and older children performed better than younger children except at larger ratios. Children’s counting proficiency was not related to success in choosing the larger of two quantities of identical-sized items. These results indicate that chimpanzees and children share an approximate number sense that is reflected through analog magnitude estimation when comparing quantities.

Paying for no reason? (Mis-)perceptions of product attributes in separate vs. joint product evaluation

Consumer decision-making involves the evaluation of options either in isolation or in relation to other alternatives present at the environment. According to Hsee’s evaluability hypothesis, it is easier to evaluate product attributes when they are juxtaposed (i.e., presented jointly) than when they are presented in isolation from each other. Recent research has provided some support to the evaluability hypothesis for the attribute of perceived product quantity. The present research tests the hypothesis in relation to the attribute of perceived fairness. In two experiments, we show that when participants evaluate products in isolation from each other, they err in their judgment of product quantity, and, consequently, they mis-attribute fairness to the seller. In a third experiment, we further show that the inclusion of constant yet unfair price information does not affect the fairness and price judgments. These findings provide evidence for the psychological plausibility of the evaluability hypothesis for the attributes of fairness and product quantity. Moreover, they suggest that isolated product evaluation may be systematically suboptimal for consumers, even when pricing information is included. Therefore, effective consumer decision-making will benefit by allowing the joint evaluation of alternatives.

Relative quantity judgments in South American sea lions (Otaria flavescens)

There is accumulating evidence that a variety of species possess quantitative abilities although their cognitive substrate is still unclear. This study is the first to investigate whether sea lions (Otaria flavescens), in the absence of training, are able to assess and select the larger of two sets of quantities. In Experiment 1, the two sets of quantities were presented simultaneously as whole sets, that is, the subjects could compare them directly. In Experiment 2, the two sets of quantities were presented item-by-item, and the totality of items was never visually available at the time of choice. For each type of presentation, we analysed the effect of the ratio between quantities, the difference between quantities and the total number of items presented. The results showed that (1) sea lions can make relative quantity judgments successfully and (2) there is a predominant influence of the ratio between quantities on the subjects' performance. The latter supports the idea that an analogue representational mechanism is responsible for sea lions' relative quantities judgments. These findings are consistent with previous reports of relative quantities judgments in other species such as monkeys and apes and suggest that sea lions might share a similar mechanism to compare and represent quantities.

Can nonhuman primates use tokens to represent and sum quantities?

It is unclear whether nonhuman animals can use physical tokens to flexibly represent various quantities by combining token values. Previous studies showed that chimpanzees (Pan troglodytes) and a macaque (Macaca mulatta) were only partly successful in tests involving sets of different-looking food containers representing different food quantities, while some capuchin monkeys (Cebus apella) have shown greater success in tests involving sets of various concrete objects representing different food quantities. Some of the discrepancy in results between these studies may be attributed to the different methods used. In an effort to reconcile these discrepancies, we presented two primates species, chimpanzees and capuchin monkeys, with two token tasks. The critical test in each task involved summing the value of multiple tokens of different types to make accurate quantity judgments. We found that, using either method, individuals of both species learned to associate individual tokens with specific quantities, as well as successfully compare individual tokens to one another or to sets of visible food items. However, regardless of method, only a few individuals exhibited the capacity to sum multiple tokens of different types and then use those summed values to make an optimal response. This suggests that flexible combination of symbolic stimuli in quantity judgments tasks is within the abilities of chimpanzees and capuchins but does not characterize the majority of individuals. Furthermore, the results suggest the need to carefully examine specific methodological details that may promote or hinder such possible representation.

Evidence for direct retrieval of relative quantity information in a quantity judgment task: Decimals, integers, and the role of physical similarity.

Participants' reaction times (RTs) in numerical judgment tasks in which one must determine which of 2 numbers is greater generally follow a monotonically decreasing function of the numerical distance between the two presented numbers. Here, I present 3 experiments in which the relative influences of numerical distance and physical similarity are assessed in just such a task using integers and decimals as stimuli. The data reveal that numerical distance is the primary feature controlling participants' RTs when integers are presented. However, the physical similarity between the decade place of the standard and the probe is the primary feature controlling participants' RTs when decimals are presented. I conclude that the unique qualities of decimals do not lend themselves to share the place-coding representation of integers, thus a direct retrieval mechanism for judging the relative quantity of decimals has developed.

An algorithm for generating color scales for both categorical and ordinal coding

AbstractPrevious research has shown multihue scales to be well‐suited to code categorical features and shown lightness scales to be well‐suited to code ordinal quantities. We introduce an algorithm, Motley, that produces color scales varying in both hue and lightness, intended to be effective for both categorical and ordinal coding, allowing users to determine both absolute and relative quantities efficiently and accurately. The algorithm first determines the lightnesses of scale colors to maximize perceived lightness differences and establish the lightness ordering, generating separate search spaces for each scale position. It then selects hues by heuristic search to maximize the discriminability of the scale. It produces scales that are ordered with respect to lightness but unordered with respect to hue and thus more discriminable than typical multihue lightness scales. In an experimental evaluation on human subjects, Motley's scales enabled accurate judgments of relative quantity, with response times superior to unordered multihue scales and comparable to ordered lightness scales, and enabled accuracy and speed of judgments of absolute quantity superior to lightness scales and comparable to multihue scales. Published 2009 Wiley Periodicals, Inc., Col Res Appl, 2010.

Preschoolers’ use of count information to judge relative quantity

Abstract We examined preschool children's use of count information in making quantity judgments. Study 1 involved 35 children 3–5 years old using a balance-scale task to judge relative quantity with or without count information provided. Study 2 replicated and extended the exploration as 54 children 3–4 years old judged relative quantity in multiple counting contexts. Children who were given count information successfully used count information in quantity evaluation when visual cues were not useful. Limited experience of counting skills, strategy choice, and limited processing capacity are each discussed as potential explanations. Implications for early childhood practice and teacher education, as well as directions for future research are explored.