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Aging and Numerosity Estimation

Centre National de la Recherche Scientifique and Université de Provence, Marseille, France.

Address correspondence to Patrick Lemaire, Centre National de la Recherche Scientifique and Université de Provence, Case D, 3 Place Victor Hugo, 13331 Marseille, France. E-mail: lemaire{at}up.univ-mrs.fr

	Abstract

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In two experiments, young and older participants were asked to find the approximate number of dots in collections including between 40 and 460 dots. Experiment 1 showed that both age groups had comparable performance and no age-related differences in the power-function exponents for numerosity. Experiment 2 found that these age-related similarities were not due to speed–accuracy trade-offs or to compensation by older adults for potential age-related decline in numerosity estimation processes. Furthermore, young and older participants' estimation performance was influenced by physical features of stimuli only for very large numerosities, presumably because these are poorly represented in long-term memory. Implications of these findings for the further understanding of how participants accomplish numerosity estimation tasks and effects of aging in this domain are discussed.

How do we estimate numerosities of large sets of items? The present two experiments document numerosity estimation (i.e., finding the approximate number of elements in sets of items) in young and older adults. In them, we attempt to determine whether estimation performance is influenced by physical features (e.g., size of items or filled area of items) for all numerosities, as suggested by many previous results, or whether the influence of physical attributes varies with numerosities. Moreover, in this study we examine the effects of aging in numerosity estimation. Before outlining the logic of the present experiments, we review previous findings on numerosity estimation.

The estimation of numerosities of large sets of items has been studied in tasks in which participants are presented collections of dots on a computer screen and are asked to provide a quick estimate of the number of dots for each collection (e.g., Beran, Smith, Redford, & Washburn, 2006). Researchers can also investigate this skill by asking participants to compare two collections of dots and to decide which is the largest (e.g., Thomas, Fowlkes, Vickery, 1980) or by asking participants to reproduce (e.g., by finger tapping) a target numerosity (e.g., Cordes, Gelman, Gallistel, & Whalen, 2001). Note that participants are not asked (or do not have time) to find the exact number of dots. Numerosity estimation is investigated in a wide variety of populations, including nonhuman animals (e.g., Beran, 2001), infants (e.g., Xue & Spelke, 2000), children (e.g., Huntley-Fenner, 2001), or adults (e.g., Boisvert, Abroms, & Roberts, 2003).

Several findings from previous studies are relevant to the present research. First, participants' estimates correlate with actual numerosity, such that they provide larger estimates with increasing numerosities. In fact, their estimates are a direct power function of the number of items presented. Indeed, estimates are reliably predicted with a power function of actual numerosity of the form E = kN^b, where E is the estimated numerosity, N is the correct numerosity, b is the power-function exponent, and k is a constant. The power-function exponents found in diverse studies are in the 0.70–0.90 range (e.g., Bevan, Maier, & Helson, 1963; Dehaene, 1997; Krueger, 1972, 1982, 1984). A power-function exponent smaller than 1 indicates that participants tend to underestimate (i.e., estimates are smaller than correct numerosities), and this underestimation increases with increasing numerosities (e.g., Krueger, 1972, 1982, 1984; Whalen, Gallistel, & Gelman, 1999). As discussed by several authors, this phenomenon is consistent with the hypothesis that memory representations for numerosities are becoming less precise and harder to discriminate with increasing numerosities (e.g., Dehaene 1997; Siegler & Opfer, 2003; Siegler & Booth, 2004).

The second important finding from previous research on numerosity estimation concerns the influence of physical features (e.g., size or arrangement of items, of stimulus display, of filled area) on participants' estimates. For example, when dots are arranged in regular patterns (e.g., as a circle or a rectangle), participants tend to provide larger estimates than they do when dots are randomly displayed (e.g., Frith & Frith, 1972; Ginsburg, 1978, 1980; Ginsburg & Pringle, 1988; Massaro, 1976). Furthermore, participants provide larger estimates for one large cluster of dots than they do for several small clusters (e.g., Ginsburg, 1991; Vos, van Oeffelen, Tibosch, & Allik, 1988) or when items are spread out than when they are bunched together (e.g., Bevan et al., 1963; Clearfield & Mix, 2001; Dixon, 1978; Ginsburg & Nicholls, 1988; Krueger, 1972; Mix, Huttenlocher, & Levine, 1996; Mix, Levine, & Huttenlocher, 1997). Participants' judgment of approximate numerosity is also highly influenced by density and texture patterns of large visual arrays (e.g., Compton & Logan, 1993; Durgin, 1995). These findings led researchers to propose that people use physical features to find estimates for all numerosities, such as the area of the stimulus field apparently occupied by a collection of dots (e.g., Allik & Tuulmets, 1991; Vos et al.).

Previous studies had two limits that we address in the present two experiments. First, to our knowledge, none of the studies directly examined the interaction between numerosities and physical properties of stimuli. Such an interaction would be observed if, for example, the same number of dots distributed differently in an array or the same number of dots with different sizes appear to contain different numbers of elements. Observing such interactions (e.g., Ginsburg & Nicholls, 1988; Krueger, 1972; Taves, 1941) would be crucial for theories of numerosity estimation, as it would no longer be possible to assume that physical features influence all numerosities when we accomplish numerosity estimation tasks. Rather, this would suggest that physical properties of stimulus influence the estimation of a subset of numerosities only. Here, we tested the hypothesis that physical properties of stimuli influence estimation performance for their numerosities that are not well represented in memory.

Although no studies directly tested the interaction between physical properties of stimulus and numerosities, closely looking at existing data sets suggests that this is a real possibility. For example, in data collected by Ginsburg and Nicholls (1988), the effects of the dot size seemed larger for large than for small numerosities. The differences in accuracy of estimates for small- and large-dot collections were larger for large numerosities than for small numerosities (also see Ginsburg, 1978, for larger regular–random differences when participants estimated small numerosities, compared with large numerosities). One of our goals in the present study was to directly test this Numerosity x Physical Attributes of Stimuli interaction on participants' estimates. We predicted that the effects of stimulus attributes would be larger for large than for small numerosities.

The second limit of previous findings is that all the studies tested only young adults. Therefore, we do not know whether numerosity estimation skills decline with age and, if so, why. Given general cognitive declines with age (see Craik & Salthouse, 2000, for an overview), we tested the possibility that older adults would provide less accurate estimates than young adults would. Alternatively, consistent with some data showing that numerical cognition is one of the cognitive areas in which aging effects are mixed (with some domains, such as complex arithmetic, showing age-related decrease and others, such as counting, showing age invariance; see Duverne & Lemaire, 2005, for an overview), we tested the possibility that young and older adults obtain comparable numerosity estimation performance. Experiment 1 compared the accuracy of estimates as well as memory representations for large numerosities in young and older adults, and Experiment 2 compared the accuracy of estimates, as well as solution latencies and eye movements.

	EXPERIMENT 1

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Experiment 1 had two goals. First, we asked whether young and older adults have a different performance in numerosity estimation tasks. Second, we aimed at determining whether memory representations for numerosities vary with age in adults. Young and older adults were asked to provide estimates of collections of dots varying in numerosities from 40 to 460 without enumerating them. The hypothesis that cognitive aging leads to decreased skills with age in numerosity estimation predicts that there will be less accurate estimates in older than in young adults. Moreover, we expected that predicting estimates as a function of correct numerosities should yield different functions in young and older adults, or similar functions with different parameters. From previous works, young adults are expected to show a power function. The hypothesis that aging is associated with different memory representations for numerosities predicts that older adults show either a different power-function exponent for numerosity or a different function.

	METHODS

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Participants
Participants were 96 individuals: 48 young adults (25 women and 23 men) with a mean age of 26.0 years (range = 24–32) and 48 older adults (27 women and 21 men) with a mean age of 73.7 years (range = 67–81). Young adults were undergraduate students from the University of Provence (Aix-en-Provence, France) who received course credit for their participation; older adults were recruited from the community and received a book on cognitive aging (Lemaire & Bherer, 2005) for their participation. All older adults had scores larger than 27 (M = 29.1) on the Mini-Mental State Examination (MMSE; Folstein, Folstein, & McHugh, 1975); therefore, none were excluded. At the end of the experiment, participants completed both the addition and the subtraction-multiplication subtests of the French Kit (French, Ekstrom, & Price, 1963), which provided assessment of participants' arithmetic fluency with an independent, paper-and-pencil test. Each subtest consisted of two pages of problems. All participants were given 2 minutes per page and were instructed to solve the problems as fast and accurately as possible. We summed the number of correct answers on each of the addition and the subtraction-multiplication tests to yield a total arithmetic score. Next, to test their verbal ability, we had participants complete the French version of the Mill Hill Vocabulary Scale (MHVS; Deltour, 1993; Raven, 1951). The MHVS consists of 33 items distributed across three pages. Each item was a target word followed by six proposed words, and the task consisted of identifying which of the proposed words had the same meaning as the target word. The number of correct items represented the level of verbal ability. Participants' characteristics are summarized in Table 1.

Procedure
We had participants tested individually in one session that lasted approximately 30 to 40 minutes. They first performed the numerosity estimation task, and then the paper-and-pencil tasks (i.e., MHVS, arithmetic fluency). At the end of the session, healthy older adults also completed the MMSE.

The experiment was controlled by E-Prime software, and stimuli were displayed on a 14-in. (36-cm) computer (Sony G-FX201 PC) screen. The program generated the displays and recorded latencies to the nearest millisecond. The display resolution was 800 x 600 pixels. Each trial was preceded by a blank screen (1,000 ms) and a fixation point (an asterisk) in the center of the screen for 750 ms. The dot patterns were then displayed in the center of the screen until participants responded (and for a maximum duration of 6 seconds). An experimenter instructed the participants to try to find the approximate number of dots in each stimulus and to report this estimate orally as soon as possible after they found an estimate. The experimenter typed in the participants' estimates. The presentation of the stimuli was random for each participant in two blocks of 42 items each, with a break of a few minutes between blocks.

	RESULTS

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In our first analysis we examined age-related differences in accuracy of estimates. Following previous studies on estimation, to measure estimation accuracy, for each problem we calculated each participant's percentage of absolute error:<--?1-->

To illustrate, suppose a participant gave 129 as an estimate for 138 dots. That participant would be 6.5% ([(129–138)/139] x 100) away from the exact numerosity. An analysis of variance (ANOVA) on each participant's mean percentage of absolute error indicates no effect of age on accuracy: Means were 31% (range = 23–39%) and 33% (range = 22–40%), F(1, 94) = 0.69, for young and older adults, respectively.

In our next analysis we examined age-related differences in the power-function exponent for numerosity. First, we fit the power function to the mean estimates for each age group. That is, we calculated the mean estimate for each numerosity generated by participants in each age group (Figure 1). Then, we predicted each group's estimate with a power function of actual numerosity, E = kN^b, where E is the estimated numerosity, N is the correct numerosity, b is the power-function exponent, and k is a constant. The power functions were comparable in young (estimate = 2.45N^.80; R² =.97) and older (estimate = 2.81N^.77; R² =.95) adults. Consistent with previous research, in our research we found that power-function exponents smaller than 1 resulted from the situation in which both young and older participants underestimated numerosities. To test for group differences in the power-function parameters, we ran individual regression analyses predicting each participant's estimate from correct numerosity. One-way ANOVAs showed no age-related differences in the power-function exponents or intercepts. Mean exponents were 0.90 and 0.93, F(1, 94) = 0.70, ns, in young and older adults respectively; corresponding mean intercepts were 0.31 and 0.27, F(1, 94) = 0.36, p =.58.

View larger version (16K): [in this window] [in a new window]   Figure 1. Mean estimated numerosities in young and older adults (Experiment 1).<--?3-->

	EXPERIMENT 2

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Experiment 2 had two goals. First, we wanted to determine if effects of physical features (i.e., size of dots) are found for all numerosities or only for a subset of numerosities. Second, we wanted to further understand age-related similarities in numerosity estimation found in Experiment 1. To achieve these ends, we analyzed the accuracy of estimates, solution times, and eye movements in young and older adults who were asked to find an approximate numerosity for small versus large collections of dots. We also manipulated the size of dots, with half of the collections including small dots and the other half including large dots.

The hypothesis that the physical features of dots influence all items predicts that the size of the dots should affect participant's estimates whatever the size of numerosities. Alternatively, the hypothesis that the physical features influence only numerosities that are hardest to estimate predicts that the size of the dots should influence large numerosities only (and not small numerosities). That is, we tested a Numerosity x Size of Dots interaction on participants' estimates. Such an interaction is possible if participants can retrieve numerosities for small collections of items better than they can for large collections. This would result from more clear and distinguishable memory representations for small than for large numerosities.

The hypothesis that comparable young and older adults' performance in numerosity estimation stems from compensation by older adults for age-related declines in cognitive resources makes predictions on patterns of solution latencies and eye movements. First, older participants may take more time than young adults to provide their estimates, and even more time for large numerosities (i.e., Age x Numerosity). Second, young and older adults may have different patterns of eye movements. For example, older adults may make more and shorter eye fixations so as to fixate distinctive portions of stimulus and thus maximize information gain with each fixation. Such a possibility would result from older adults' reduced useful field of view (Scialfa, Kline, & Lyman, 1987; Sekuler, Bennet, & Mamelak, 2000; Watson, Maylor, & Bruce, 2005). More importantly, the Age x Numerosity interaction on eye movements should show different effects of numerosities on young and older adults' mean number and duration of fixations, amplitude of saccades, and dispersions of regard (or breadth of visual scanning of stimuli). Like for the corresponding Age x Numerosity interaction on latencies, this can happen if memory representations for large numerosities are not as clear and distinguishable as those for small numerosities, and even more so in older adults. Finally, Experiment 2 offered us the possibility of testing the Age x Numerosity x Size of Dots interaction. Such an interaction is possible if the Numerosity x Size of Dots interaction is significant in young adults but not in older adults, such that the effect of numerosity is significant in young adults only while they are estimating large numerosities. Alternatively, similar Numerosity x Size of Dots interactions in both age groups would be additional evidence of age invariance in numerosity estimation.

	METHODS

TOP Abstract Experiment 1 Methods Results Experiment 2 Methods Results General Discussion References

 
Participants
Participants were 54 individuals: 27 young adults (17 women and 10 men) with a mean age of 25.6 years (range = 22–30) and 27 older adults (20 women and 7 men) with a mean age of 70.9 years (range = 65–88). Young adults were undergraduate students from the University of Provence (Marseille, France) who received course credit for their participation; older adults were recruited from the community and received a book on cognitive aging (Lemaire & Bherer, 2005) as a way to thank them for their participation. As in Experiment 1, in Experiment 2 we assessed participants' verbal and arithmetic fluency, and all older adults took the MMSE. Participants' characteristics are summarized in Table 1.

Stimuli
Each participant solved 128 numerosity estimation problems. Each stimulus was made of black dots randomly displayed on a white screen. We factorially manipulated two types of problem features, namely size of numerosity (i.e., small vs large numerosity) and dot size (i.e., small vs large dots). Half the small-dot problems had dots with 14 pixels and the other half had dots with 20 pixels; half the large-dot problems had dots with 38 pixels and the other half had dots with 44 pixels. Moreover, half the small-numerosity problems included 49 dots and the other half included 78 dots; half the large-numerosity problems included 91 dots and the other half included 147 dots. To control for potential factors such as filled area or between-dot distances, we had all dots occupy a small area (100 cm²; 250 x 250 pixels) in the center of the computer screen for half the stimuli and a large area (289 cm²; 430 x 430 pixels) for the other stimuli. Within each stimulus, all dots had the same size and two adjacent dots were separated by at least one pixel so that no pixels from different dots overlapped. As participants sat approximately 60 cm from the screen, stimuli occupied 8.6° or 15.8° of visual angle for stimuli occupying a small or large surface, respectively.

Procedure
The procedure was the same as in Experiment 1, except that we had an experimenter record the participants' solution latencies and eye movements during the experiment by using an iView X Eyetracking Device (SensoMotoric Instruments, Berlin, Germany). The experimenter asked participants not to make too much head or body movement and helped them with the chin rest of the iView X system. We had the experimenter perform the calibration by asking participants to view nine crosses on the screen. Recalibration was performed between each block if necessary. We had eye position sampled every 20 ms (at a sampling rate of 50 Hz) and analyzed it offline by using customized software. A timer was started when collections of dots appeared on the screen and ended when the experimenter pressed on the space bar of the computer keyboard, which happened as soon as possible after participants provided their oral response. We used this procedure because pilot testing showed that when response time was recorded from participants' first vocalization, they frequently changed their answer or were still estimating during the production of the response. Experimenter timing of responses also minimized response demands on participants and potential loss of trials as a result of voice key artefacts. We had participants tested individually in one single session that lasted approximately 60 minutes.

	RESULTS

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We report the results in two main parts. The first analyzes approximate quantification performance; the second looks at patterns of eye movements. In all results, unless otherwise noted, differences are significant to at least p <.01.

Approximate Quantification Performance
We analyzed the mean solution times and percentages of deviation with 2 (Age: young, older adults) x 2 (Numerosity: small, large) x 2 (Size of Dots: small, large) ANOVA designs, with age as the only between-subjects factor (see Table 2).

ANOVAs on solution latencies revealed significant main effects of numerosity, F(1, 52) = 7.23, MSE = 201439, p =.01, ² =.09, showing that participants were faster at estimating small numerosities (3,832 ms) than large numerosities (3,996 ms). Moreover, the Numerosity x Size of Dot interaction was significant, F(1, 52) = 16.70, MSE = 63,573, ² =.32. This interaction occurred because participants were faster with large dots (3,733 ms) than with small dots (3,931 ms) while they were estimating small numerosities, but equally fast with small- and large-dot collections (3,955 ms and 4,037 ms) when they were estimating large numerosities. The main effect of age was not significant, F(1, 52) = 1.41, p =.24, MSE = 28,781,676, although young adults tended to be faster than older adults (3,480 ms vs 4,348 ms). However, age interacted with dot size, F(1, 52) = 6.96, MSE = 65,145, p =.01, ² =.17. Young adults were not influenced by dot size, that is, M = 3,463 ms and M = 3,497 ms, F < 1, for small and large dots, respectively; however, older adults were influenced, that is, M = 4,423 ms and M = 4,273 ms, F(1, 52) = 10.9, MSE = 55,289, for small and large dots, respectively. No other effects came out significant on solution times or percent deviations.

Eye Movement Data
We analyzed the mean number and duration of fixations, mean amplitude of saccades, and dispersion of regard (DOR) with the following ANOVA designs: 2 (Age) x 2 (Numerosity: small, large) x 2 (Size of dots: small, large), with age as the only between-subjects factor.

Mean number, mean duration of fixations, and mean amplitude of saccades
Age was the only factor that significantly influenced mean number and duration of fixations. Young adults made fewer fixations than older adults did (7.4 vs 9.1); F(1, 52) = 4.20, p =.02, MSE = 54.96, ² =.06. They also made longer fixations (420 ms vs 343 ms); F(1, 52) = 6.03, p =.01, MSE = 52,502, ² =.10. Analyses of mean amplitudes of saccades showed no significant main or interaction effects (Fs < 2.14). Young and older participants had equally large saccades (4.5° vs 4.6°), and both age groups made saccades of comparable amplitudes on small- and large-dot collections (4.6° vs 4.5°) or on small and large numerosities (4.3° vs 4.7°).

Dispersion of regard
To investigate the spatial distribution of attention during each trial, we calculated the DOR on the grid during approximation. This DOR was the mean of standard deviations of points of regard in the X and Y axes. The 0, 0 coordinates on the X, Y axis arbitrarily referred to the upper left corner of the screen and the 800, 600 coordinates to the lower right corner. Larger DOR means that participants visually scanned more surface area of stimulus, independently of saccade amplitudes. Older participants scanned larger surface of stimuli than young adults did (59.7 vs 52.9); F(1, 52) = 4.65, p =.02, MSE = 929.81, ² =.08. When participants scanned large numerosities, they scanned stimuli with large dots less broadly than stimuli with small dots (54.8 vs 57.6), F(1, 52) = 18.99, ² =.27, but they scanned stimuli with large and small dots (56.1 vs 56.7, F < 1) similarly while they were estimating small numerosities. No other effects came out significant on DOR.

	GENERAL DISCUSSION

TOP Abstract Experiment 1 Methods Results Experiment 2 Methods Results General Discussion References

 
We found the following phenomena that are important for understanding how young and older people find approximate numerosities of large sets of items. Participants' estimates were influenced by both the number of dots in a collection and the size of dots themselves. There were no age-related differences. Finally, young adults made fewer and longer eye fixations and scanned stimuli less broadly than older adults did. In this section, we discuss the implications of these findings to further understand numerosity estimation and effects of aging in this activity.

The present results replicate and complement previous findings regarding effects of numerosities and physical features in numerosity estimation. First, participants' estimates could be predicted from actual numerosity with a power function. Here, as in previous studies, the power-function exponents in numerosity estimation were in the 0.70–0.90 range (e.g., Krueger, 1984). Such smaller-than-one exponents indicate that participants underestimate the number of elements in large collections of items. In this study, this underestimation phenomenon was generalized to very large numerosities. Moreover, the present study replicated effects dot size, first found by Ginsburg and Nicholls (1988) for numerosities smaller than those tested here.

To our knowledge, the present study is the first to find that the effect of dot size is restricted to large numerosities. Dot size exerted no effects on participants' estimates of small numerosities. Presumably, participants have better, clearer, and more easily distinguishable memory representations for small than for large numerosities. Such better memory representations led participants to be influenced by physical features to a much lower extent than on the large, most poorly represented numerosities in memory. At a more general level, this suggests that physical features interact with internal representations of numerosities (see Kotary & Hoyer, 1995, for a similar suggestion in enumeration tasks). Future studies may test whether such a conclusion holds to estimation activities other than numerosity estimation (Dixon, 1978). Estimating dimensions of stimuli such as weight, length, or distance are often influenced by another, irrelevant though sometimes correlated, dimension (e.g., Krueger, 1984). For example, participants are influenced by the volume of objects when they estimate their weight, estimating as heavy those objects that have big volume (e.g., Nyssen & Bourdon, 1956).

One of the most interesting results in the present study concerns age invariance in participants' performance. Both accuracy of estimates and solution latencies were comparable in young and older adults, and there were no age-related differences in the power-function exponents for numerosity. Most interesting was the fact that the effects of dot size interacted with numerosities in both young and older adults. Such age invariance in numerosity estimation is interesting, as it adds further evidence that some cognitive activities are spared with age in contrast to other age-impaired cognitive activities. One possibility is that numerosity estimation involves automatic visual-spatial or perceptual processing (e.g., encoding of patterns of dots) that requires little or no central attention. In several previous studies, age-comparable performance has been found in cognitive tasks that require little central attention. Examples include simple arithmetic (see Duverne & Lemaire, 2005, for a review), psychological-refractory-period effects (e.g., Allen, Lien, Murphy, Sanders, & McCann, 2002), visual word recognition (e.g., Lien et al., 2006), or detection of multiple features in visual search tasks (e.g., Bucur, Allen, Sanders, Ruthruff, & Murphy, 2005). It is also possible that numerosity estimation tasks involve specific cognitive processes that are age invariant. This could arise as numerosity estimation might rely on a phylogenetically old system of numerical representations. This system rests greatly on the manipulation of mental magnitudes or nonverbal representations of approximate number and is often viewed as a prerequisite for symbolic mathematical processing (e.g., Dehaene, 1997, Gallistel & Gelman, 1992, 2000; Geary & Lin, 1998; Pica, Lemer, Izard, & Dehaene, 2004). Such a presymbolic system might suffer less from age-related declines than do other cognitive systems.

Before accepting the conclusion that aging does not influence numerosity estimation, we find it important to consider two points. The present age equivalence may be the result of cohort effects. Moreover, numerosity estimation skills may decline with age, but older adults did use compensatory mechanisms to circumvent deleterious effects of age.

It is impossible to discard the hypothesis that the present age equivalence in numerosity estimation is a cohort effect. As found in other domains of cognition, such as arithmetic (Geary, Salthouse, Chen, & Fan, 1996), it is possible that the age equivalence found in numerosity estimation stems from older adults' quantification skills having been well developed at school or practiced during their adults' daily lives. Such practice or training effects would yield highly functional numerosity estimation processes in older adults. Using cross-sequential or cross-cultural comparisons (e.g., those by Geary et al., 1996) will enable one to determine if this activity is truly age invariant during adulthood.

The different patterns of eye movements across age groups (i.e., shorter and more numerous fixations together with larger visual scanning of stimuli in older adults than in younger ones) would be consistent with the hypothesis that older adults used different numerosity estimation strategies. Such strategies would enable older adults to compensate for age-related declines in numerosity estimation. However, four points suggest that this might not be the case. First, differences in eye movements may result from older adults' reduced useful field of view (e.g., Ball et al., 1988; Scialfa et al., 1987; Sekuler et al., 2000; Watson et al., 2005). By doing more fixations and fixating more shortly than young adults, older adults may have tried to maximize how much information they could encode in each gaze, which is a reasonable encoding strategy to adopt given a less efficient visual system. Second, the lack of interactions between age, numerosity, and dot size and between age and numerosity on eye movements suggests that this age-related difference in encoding strategies reflects a less efficient visual system rather than degraded memory representations for numerosities. Indeed, the latter predicts that older adults would have greater difficulties than young adults to do numerosity estimation tasks, especially when quantifying large collections of dots. This was found here in none of our measures. Third, the effect of dot size on older adults' solution times, and lack thereof in young adults, is consistent with an encoding deficit account. Large dots helped older adults to more quickly encode stimuli. Finally, assessments of the specific strategies (e.g., by means of verbal protocols) that young and older participants use on each problem are needed to directly assess the possibility of compensatory mechanisms in older adults.

To conclude, the present study showed that participants' performance on numerosity estimation resulted from the participants' use of both semantic (numerosity size) and physical (dot size) features of stimuli, and that these two features interact to support people's performance. Moreover, we found no age-related declines in numerosity estimation, above and beyond peripheral encoding differences; this finding suggests that estimation skills may be one of the rare cognitive domains that show no age-related decline.

	Acknowledgments

 
This research was supported by Grant 06-2-235061 from the Agence Nationale de la Recherche. We thank Stéphane Dufau for writing the Matlab program to generate stimuli, and Cynthia Cambours and Lorane Pernet for their help in collecting the data.

	Footnotes

 
Decision Editor: Thomas M. Hess, PhD

Received for publication December 22, 2006. Accepted for publication June 1, 2007.

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