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3 Individual differences in the components4 of children’s and adults’ information processing5 for simple symbolic and non-symbolic numeric6 decisions

7

8

9 Clarissa A. Thompson a,⇑, Roger Ratcliff b, Gail McKoon b

10 aDepartment of Psychological Sciences, Kent State University, Kent, OH 44242, USA11 bDepartment of Psychology, The Ohio State University, Columbus, OH 43210, USA

1213

1 5a r t i c l e i n f o

16 Article history:17 Received 17 November 201518 Revised 18 April 201619 Available online xxxx

20 Keywords:21 Diffusion model22 Response time23 Accuracy24 Cognitive development25 Numerical ability26 Individual differences27

2 8a b s t r a c t

29How do speed and accuracy trade off, and what components of30information processing develop as children and adults make simple31numeric comparisons? Data from symbolic and non-symbolic num-32ber tasks were collected from 19 first graders (Mage = 7.12 years), 2633second/third graders (Mage = 8.20 years), 27 fourth/fifth graders34(Mage = 10.46 years), and 19 seventh/eighth graders (Mage = 13.2235years). The non-symbolic task asked children to decide whether36an array of asterisks had a larger or smaller number than 50, and37the symbolic task asked whether a two-digit number was greater38than or less than 50. We used a diffusionmodel analysis to estimate39components of processing in tasks from accuracy, correct and error40response times, and response time (RT) distributions. Participants41who were accurate on one task were accurate on the other task,42and participants who made fast decisions on one task made fast43decisions on the other task. Older participants extracted a higher44quality of information from the stimulus arrays, were more willing45to make a decision, and were faster at encoding, transforming the46stimulus representation, and executing their responses. Individual47participants’ accuracy and RTs were uncorrelated. Drift rate and48boundary settings were significantly related across tasks, but they49were unrelated to each other. Accuracy was mainly determined by50drift rate, and RT was mainly determined by boundary separation.51We concluded that RT and accuracy operate largely independently.52� 2016 Elsevier Inc. All rights reserved.53

http://dx.doi.org/10.1016/j.jecp.2016.04.0050022-0965/� 2016 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (C.A. Thompson).

Journal of Experimental Child Psychology xxx (2016) xxx–xxx

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Please cite this article in press as: Thompson, C. A., et al. Individual differences in the components of children’s andadults’ information processing for simple symbolic and non-symbolic numeric decisions. Journal of ExperimentalChild Psychology (2016), http://dx.doi.org/10.1016/j.jecp.2016.04.005

http://dx.doi.org/10.1016/j.jecp.2016.04.005

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5455 Introduction

56 One debate in the numerical cognition literature focuses on whether performance across individ-57 uals on precise symbolic numerical tasks, in which children and adults compare Arabic numerals such58 as 45 versus 50, is correlated with performance on approximate non-symbolic numerosity tasks, in59 which participants compare 45 versus 50 dots. The nature of the relation between these tasks and60 among these tasks and overall mathematics achievement tests is unclear because there is a discrep-61 ancy in the findings reported in the numerical cognition literature. Sometimes performance on sym-62 bolic and non-symbolic tasks is correlated, but sometimes it is not (De Smedt, Verschaffel, &63 Ghesquiere, 2009; Fazio, Bailey, Thompson, & Siegler, 2014; Gilmore, Attridge, & Inglis, 2011;64 Halberda, Ly, Wilmer, Naiman, & Germine, 2012; Holloway & Ansari, 2009; Maloney, Risko, Preston,65 Ansari, & Fugelsang, 2010; Price, Palmer, Battista, & Ansari, 2012; Sasanguie, Defever, Van den66 Bussche, & Reynvoet, 2011). Sometimes performance on non-symbolic comparison tasks is correlated67 with mathematics achievement scores, and sometimes it is not (De Smedt et al., 2009; Durand, Hulme,68 Larkin, & Snowling, 2005; Fazio et al., 2014; Gilmore, McCarthy, & Spelke, 2010; Halberda, Mazzocco,69 & Feigenson, 2008; Holloway & Ansari, 2009; Inglis, Attridge, Batchelor, & Gilmore, 2011; Libertus,70 Feigenson, & Halberda, 2011; Lyons & Beilock, 2011; Mazzocco, Feigenson, & Halberda, 2011a,71 2011b; Mundy & Gilmore, 2009; Price et al., 2012).72 Recent research by Inglis and colleagues (2011), Halberda and colleagues (2012), and Fazio and73 colleagues (2014) illustrates variability in the reported strength of the relationship between74 non-symbolic tasks and mathematics achievement. Inglis and colleagues (2011) reported a strong75 correlation for 7- to 9-year-old children between accuracy on the calculation subtests of the Wood-76 cock–Johnson mathematics achievement test and accuracy on a non-symbolic dot comparison task.77 Accuracy was operationalized as the Weber fraction, w, which is a measure of precision/acuity of an78 individual’s approximate number system (ANS). It is hypothesized that the ANS allows humans of79 all ages, as well as animals, to approximate numerical magnitudes or values (Dehaene, Dehaene-80 Lambertz, & Cohen, 1998). Larger values of w indicate lower levels of numerical precision. Inglis81 and colleagues found a negative correlation between w on the non-symbolic comparison task and82 standardized mathematics achievement scores for children but not for adults. That is, those children83 who had larger Weber fractions had lower standardized mathematics achievement scores. Inglis and84 colleagues’ (2011) findings contrast with those of Halberda and colleagues (2012), who used a large85 internet-based sample of children and adults between 11 and 85 years of age to complete a non-86 symbolic number comparison task. Halberda and colleagues found a small but significant correlation87 between ANS precision on the non-symbolic comparison task and school mathematics ability across88 all ages tested. Fazio and colleagues (2014) found that after controlling for standardized reading per-89 formance, a non-mathematics test of basic cognitive ability, both symbolic and non-symbolic perfor-90 mance uniquely predicted mathematics achievement scores. That is, both symbolic knowledge and91 non-symbolic knowledge contribute to overall mathematics achievement. However, performance on92 symbolic tasks explained a greater proportion of the variance in mathematics achievement than did93 performance on non-symbolic tasks. Taken together, these three examples illustrate the variability94 in the strength of the correlations that exist between non-symbolic task performance and mathemat-95 ics achievement across the lifespan. Two recent meta-analyses (Chen & Li, 2014; Fazio et al., 2014)96 provided possible explanations for these variable findings.97 Chen and Li’s (2014) meta-analysis claims that one source of the discrepant results is that many of98 the previously conducted studies may have been underpowered, and the small sample sizes led to99 spurious conclusions. Fazio and colleagues’ (2014) meta-analysis found a small reliable correlation

100 between non-symbolic numerical performance and overall mathematics achievement. The size of101 the correlations reported in the individual studies that comprised the meta-analysis differed with102 regard to the dependent variables that the studies’ authors chose. For example, when accuracy (per-103 centage correct) or accuracy-based measures (w) in addition to response time (RT) was the dependent104 variable of choice, the correlation between performance on the non-symbolic numerical task and stan-105 dardized mathematics achievement was stronger than when a measure of RT was used alone. The

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106 relationship between performance on non-symbolic tasks and overall mathematics achievement was107 also stronger for children under 6 years of age as compared with children between 6 and 18 years of108 age and adult participants. This finding is consistent with age differences found by Inglis and109 colleagues (2011). Fazio and colleagues interpreted the age differences to mean that formal schooling110 might play a role in the developmental differences and offered possible interpretations. First, children111 under 6 years of age might be more reliant on non-symbolic numerical representations than children112 in the 6- to 18-year-old age range and adults, who presumably have extensive experience in working113 with symbolic numbers in formal schooling environments. Second, mathematics achievement tests for114 children under 6 years of age might be more sensitive to non-symbolic numerical knowledge. In sum-115 mary, Fazio and colleagues’ (2014) meta-analysis indicated that the age of the participants completing116 the non-symbolic and mathematics achievement tests and the dependent variables chosen to measure117 the numerical knowledge might contribute to the discrepancies found in the numerical cognition118 literature.119 Recent work by Ratcliff, Thompson, and McKoon (2015), in which the diffusion model (Ratcliff,120 1978) was applied to adults’ performance on symbolic and non-symbolic discrimination tasks, agreed121 with conclusions drawn by Fazio and colleagues (2014). They argued that it is misleading for research-122 ers to report only RT or accuracy from symbolic and non-symbolic discrimination tasks because each123 dependent variable could present a slightly different view of participants’ numerical information pro-124 cessing abilities. Intuition might suggest that RT and accuracy are negatively correlated with one125 another such that better performance means more accurate responses and faster responses relative126 to the average values for the group (e.g., see Siegler’s (1996) overlapping waves theory that describes127 how children know and use various strategies that differ in accuracy and efficiency). However, Ratcliff128 and colleagues (2015) provided evidence that an individual’s accuracy and RT were not correlated129 within symbolic and non-symbolic numerical discrimination tasks (this is also true in other tasks;130 Ratcliff, Thapar, & McKoon, 2010, 2011). Knowing how accurately a participant decided whether 45131 was greater than or less than 50 did not predict how quickly the participant made the decision. The132 diffusion model, described in detail below, provides an explanation of this result. The model was fit133 to correct and error RT distributions and accuracy, and it produced model parameters representing134 the quality of evidence (drift rate), the amount of evidence needed for a decision (boundary separa-135 tion), and the duration of non-decision processes (non-decision time). Application of the model136 showed that boundary settings and drift rates were largely uncorrelated with each other and that137 accuracy was mainly determined by drift rate and RT was largely determined by boundary separation138 (and non-decision time). This provided a way of understanding the lack of correlation between RT and139 accuracy (but did not predict why there was no correlation). People use global boundary settings for140 decisions, and they do not modulate their settings as a function of whether they are good or bad at a141 particular task (Ratcliff et al., 2015).142 In this article, we show how, according to the diffusion model, accuracy and RT and dependent143 measures based on them do not assess independent abilities but instead are manifestations of the144 same underlying cognitive processes. In the studies cited above (Fazio et al., 2014; Halberda et al.,145 2012; Inglis et al., 2011; Ratcliff et al., 2015), the response required of a participant is a decision146 between two (or more) alternatives (e.g., ‘‘is this number larger or smaller than 50?”). Whatever147 the quality of a participant’s numerosity information, a response must be chosen, and the choice will148 take some amount of time. Accuracy and speed can trade off, and the trade-off is under a participant’s149 control. A participant might decide to respond as quickly as possible, sacrificing accuracy, or as accu-150 rately as possible, sacrificing speed. If a participant adopts a speed emphasis, the slope of the RT–dif-151 ficulty function will be lower than if the participant adopts an accuracy emphasis (Ratcliff et al., 2015,152 Experiment 2). In consequence, differences among participants in the quality of the numeracy infor-153 mation on which they base their decisions can be obscured by differences in their speed/accuracy set-154 tings. The only way to separate information quality from speed/accuracy settings is to understand how155 they interact. We do that with a sequential sampling decision model, Ratcliff’s (1978) diffusion model156 (see also Ratcliff & McKoon, 2008), which is described below.

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157 The diffusion model

158 In Ratcliff’s (1978) diffusion model (see also Ratcliff & McKoon, 2008), there is noisy accumulation159 of stimulus information over time toward one of two boundaries. A response is made when the160 amount of accumulated information reaches one of two boundaries, or criteria, one for each of the161 two possible choices (e.g., ‘‘is 60 greater than or less than 50?”). The rate of accumulation, called ‘‘drift162 rate,” is determined by the quality of the information used in a decision. For example, information163 about ‘‘is 9 greater than 1?” would be stronger than information about ‘‘is 2 greater than 1?” and164 the information available to a high school student would likely be stronger than the information avail-165 able to a second grader (e.g., Ratcliff, Love, Thompson, & Opfer, 2012). Drift rate corresponds to numer-166 ical acuity in the numerosity and number discrimination tasks.167 Fig. 1 shows the operation of the model. Total RT is the time it takes to encode a stimulus, trans-168 form the stimulus representation to a decision-relevant representation, compare the representation169 with memory, decide on a response, execute a response, and so forth. The transformation from the170 stimulus to a decision-relevant representation maps the many dimensions of a stimulus (e.g., size,171 color, shape, number) onto the task-relevant dimension—the drift rate that drives the decision pro-172 cess. The accumulation of information begins at a starting point (z in Fig. 1) and proceeds until one173 of the two boundaries is reached (a or 0 in the figure). Because the accumulation process is noisy,174 for a given value of drift rate, at each instant of time there is some probability of moving toward175 the correct boundary and some smaller probability of moving toward the incorrect boundary. This176 variability means that accumulated information can hit the wrong boundary, producing errors, and177 that stimuli with the same values of drift rate can hit a boundary at different times. For application178 of the model, non-decision processes (e.g., stimulus encoding, transformation to task-relevant infor-179 mation, response execution) are combined into one parameter, Ter in Fig. 1. As illustrated in the figure,180 the model predicts the right-skewed shapes of RT distributions that are observed empirically in two-181 choice tasks.182 The model decomposes accuracy and RTs into the three main components just described: drift183 rates, boundary settings, and non-decision processes. The values of these components vary from trial184 to trial because, it is assumed, participants cannot accurately set identical values from trial to trial185 (e.g., Laming, 1968; Ratcliff, 1978). Across-trial variability in drift rate is assumed to be normally dis-186 tributed with SD g, across-trial variability in the starting point (equivalent to across-trial variability in

0

z

a correct correct

error

Correct RT distribution

Error RT distribution

DriftRate v

response

responseresponse

Time

A decision boundary

B decision boundary

Time

Total RT=u+d+w

u

d

range=st

mean=Ter

w

decision

encoding etc. responseoutput

Non-decisioncomponents of RT=u+w

Fig. 1. Illustration of the diffusion model. The top panel shows three simulated paths with drift rate v, starting point z, andboundary separation a. Drift rate is normally distributed with SD g, and starting point is uniformly distributed with range sz.Non-decision time is composed of encoding processes, processes that turn the stimulus representation into a decision-relatedrepresentation, and response output processes. Non-decision time has mean Ter and a uniform distribution with range st.



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187 the boundary positions) is assumed to be uniformly distributed with range sz, and across-trial variabil-188 ity in the non-decision component is assumed to be uniformly distributed with range st. These distri-189 butional assumptions are the ones usually made, but they are not critical as long as they are within190 their usual ranges (Ratcliff, 2013).191 Because the model decomposes accuracy and RTs into components and separates out variability in192 them, the power to observe effects of independent variables on performance can be substantially193 increased. For example, lexical decision experiments have been used to attempt to identify partici-194 pants with high anxiety by looking at their performance on ‘‘threat” words (e.g., anger, hostility,195 attack). Significant differences between high-anxiety and low-anxiety participants did not appear196 with RTs or accuracy but did appear with drift rates. The model analyses increased power by a factor197 of approximately 2 (White, Ratcliff, Vasey, & McKoon, 2010).198 Current theories about numeracy are constrained only by mean RTs for correct responses or only by199 accuracy. The diffusion model is more tightly constrained and can be falsified. The most powerful con-200 straint comes from the requirement that the model fit the right-skewed shape of RT distributions, as201 shown in Fig. 1 (Ratcliff, 1978, 2002; Ratcliff & McKoon, 2008; Ratcliff, Van Zandt, & McKoon, 1999).202 Ratcliff (2002) generated simulated data for which RT distributions behaved across conditions in ways203 that are plausible but never obtained empirically. In all cases, the model failed to fit the data. In addi-204 tion, across experimental conditions that vary in difficulty, such as when target numbers differ in205 absolute distance from a comparison number (e.g., 50) in a symbolic number discrimination task,206 changes in accuracy, RT distributions, and the relative speeds of correct and error responses must207 all be captured by changes in only one parameter of the model, namely drift rate. Across experimental208 conditions that vary in speed/accuracy criteria (e.g., speed vs. accuracy instructions to participants), all209 changes in accuracy, RT distributions, and the relative speeds of correct and error responses are usu-210 ally captured by changes only in the settings of the boundaries. The boundaries cannot be adjusted as211 a function of difficulty because it would be necessary for the system to know which level of difficulty212 was being tested before boundary settings could be determined. The model explains the ways in213 which drift rates and boundary settings can interact to determine participants’ RTs and accuracy.214 For a given value of drift rate, a participant can adopt wider boundaries and so be more accurate215 but slower, whereas the participant can adopt narrower boundaries and so be faster but less accurate.216 Across participants, drift rates and boundary settings can differ independently. Participants who have217 high drift rates will have good accuracy and fast responses when their boundaries are close together,218 and they will have good (perhaps slightly better) accuracy and slow responses when their boundaries219 are farther apart. Participants who have low drift rates will have poor accuracy and fast responses220 when their boundaries are close together, and they will have (perhaps) somewhat better accuracy221 and slow responses when their boundaries are far apart. To put this another way, participants with222 fast responses can be accurate or inaccurate, and participants with slow responses can be accurate223 or inaccurate (cf. Ratcliff, Thapar, & McKoon, 2006; Ratcliff et al., 2010, 2011). That boundary settings224 are under a participant’s control has been demonstrated in past studies where participants responded225 to instructions to maximize speed by decreasing the settings (Ratcliff, Thapar, & McKoon, 2001, 2003,226 2004; Thapar, Ratcliff, & McKoon, 2003). In the model, accuracy is related to drift rate and RT is related227 to speed–accuracy criteria, but drift rate and criteria are not related to each other across participants.228 This provides a theoretical basis for understanding why the common belief that accurate participants229 do not always make the quickest decisions does not always hold true. Ratcliff and colleagues (2015)230 provided evidence that accuracy, RT, and dependent measures based on them do not assess indepen-231 dent abilities but instead are manifestations of the same underlying cognitive processes. The finding232 that accuracy–RT correlations were not significant calls into question interpretations of results from233 the many previous studies in the numerical cognition literature for which only RTs or only accuracy234 values were reported. The finding also puts a stringent constraint on theories about number process-235 ing; whatever it is that determines a participant’s overall accuracy is not directly related to whatever it236 is that determines the participant’s overall speed. Any theory about performance in numeracy tasks237 must explain why this is so. In summary, the diffusion model breaks down accuracy and RT into238 components of information processing, and these resulting model parameters are not significantly239 correlated with one another. Thus, the model allows researchers to assess speed–accuracy240 relationships.



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241 The model also solves a scaling problem for RT and accuracy. In the numeracy literature, when a242 task has high accuracy the performance measure is typically RT (or a measure based on RT), and when243 a task has low accuracy the measure is typically accuracy (or a measure based on accuracy). The two244 measures have different scales; accuracy varies from chance to ceiling and asymptotes at 1, and RTs245 vary from a short minimum to potentially very long duration. The diffusion model resolves this issue246 because the two measures come from the same underlying processes.247 The model also helps to address problems with ceiling and floor effects (Ratcliff, 2014). For some248 experiments, accuracy might be at chance for several of the most difficult conditions (e.g., ‘‘is 48 more249 or less than 50?”), and it might be at ceiling for several of the easiest conditions (e.g., ‘‘is 13 more or250 less than 50?”). Model parameters (especially parameters that represent across-trial variability in251 model components) can be estimated when accuracy is not at ceiling or floor (0 or 1). When accuracy252 is at ceiling or floor, the model can estimate differences in drift rates because the other conditions253 allow model parameters to be estimated. Because RTs differ across these high-accuracy conditions,254 RTs are sufficient to determine drift rates. Ceiling effects are likely present in symbolic number dis-255 crimination tasks, especially for older children and adults, because participants have extensive expe-256 rience in thinking about and processing whole numbers.

257 The current experiment

258 Ratcliff and colleagues (2015) tested college-aged adults on four numerical tasks: (a) numerosity259 discrimination (a non-symbolic numerical decision task in which participants decided whether a stim-260 ulus array contained more or less than 50 asterisks), (b) number discrimination (a symbolic numerical261 decision task in which participants decided whether an Arabic numeral was greater than or less than262 the number 50), (c) memory for two-digit numbers (participants indicated whether two-digit num-263 bers on a test list appeared on an immediately preceding study list), and (d) memory for three-digit264 numbers (participants indicated whether three-digit numbers on a test list appeared on an immedi-265 ately preceding study list). Results provided evidence that ‘‘on-line” decision tasks, such as numerosity266 and number discrimination that require perceptual discrimination yet do not require memory access,267 were correlated with performance on ‘‘off-line” tasks that do require memory access, such as memory268 for two- and three-digit numbers. Accuracy positively correlated across tasks, as did RTs. The more269 accurate an individual was in deciding whether the number of asterisks on the computer screen270 was greater than or less than 50 during the numerosity discrimination task, the more accurate the271 individual was when deciding whether a numeral was greater than or less than 50 in the number dis-272 crimination task. The faster a participant responded on one task, the faster the participant responded273 on the other task. However, if a participant was accurate, it did not mean that the participant was fast274 (and vice versa). There were significant positive correlations across the tasks between the quality of275 the numeracy information (drift rate) driving the decision process and between the speed/accuracy276 criterion settings, suggesting that similar numeracy skills and similar speed–accuracy settings are277 involved in numerosity and number discrimination and memory for two- and three-digit numbers.278 The novelty of our current experiment is that the diffusion model can account for correct and incorrect279 RTs, RT distributions, and accuracy. The model decomposes RTs into three main information process-280 ing components, and these parameters can indicate what develops over time as children becomemore281 accurate and faster at making numerical decisions. The advantage of applying the diffusion model to282 cross-sectional data is that the mechanism of change in simple numerical decisions across the lifespan283 can be highlighted. For instance, if drift rates increase with age or grade level, it could indicate that284 improvement in children’s magnitude understanding, or numerical acuity/discrimination ability, is285 driving the decision process. If boundary separation becomes narrower with age, it could indicate that286 children become more willing to execute a response. Finally, if the non-decision component becomes287 faster with age, a model-based interpretation is that the children are becoming more skilled at trans-288 forming the stimulus array into a form that will allow them to execute a discrimination decision.289 In our current experiment, our first aimwas to replicate the findings of Ratcliff and colleagues (2015)290 with children by investigating whether elementary and middle school students’ performance on the291 non-symbolic numerosity discrimination task (‘‘is the number of asterisks in a 10 � 10 array greater292 than or less than 50 asterisks?”) was correlated with their performance on the symbolic number



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293 discrimination task (‘‘is an Arabic numeral greater than or less than the Arabic numeral 50?”).We tested294 four groups of children—first graders, second/third graders, fourth/fifth graders, and seventh/eighth gra-295 ders—and then compared the children’s performancewith performance of the adults reported in Ratcliff296 and colleagues (2015). A second aimwas to investigatewhether therewere developmental progressions297 in diffusion model parameters across the lifespan on a symbolic numerical discrimination task. Ratcliff298 and colleagues (2012) reported developmental differences on a non-symbolic numerical discrimination299 task in drift rate, boundary separation, and non-decision time across the lifespan. Children, much like300 older adults, made slower decisions than did college-aged adults. However, developmental differences301 in the diffusionmodel parameters indicated that children and older adults made slow decisions for dif-302 ferent reasons. Younger children requiredmore evidence prior tomaking a decision, setmore conserva-303 tive decision criteria, and took longer to encode stimulus information, transform the representation to304 produce a decision related to numerical quantity, and execute a decision than did older children and305 college-aged adults. It is an open question as to whether drift rate, boundary separation, and non-306 decision times change across the lifespan for other numeracy tasks such as the number discrimination307 task, a task on which participants presumably improve because they are learning more about the rela-308 tions between whole numbers during formal schooling. A third aim was to examine individual differ-309 ences in accuracy and RT between the two tasks, to examine individual differences in model310 parameters between tasks, and to examine correlations between accuracy and RTwithin each task. This311 allowed us to investigate whether RT and accuracy on symbolic and non-symbolic discrimination tasks312 were relatively independent measures of performance within individual children, as was found previ-313 ouslywith adults. If so, we could then argue against the practice of choosing one or the other dependent314 variable alone to assess children’s numerical discrimination abilities.

315 Method

316 Participants

317 Ratcliff and colleagues (2015) tested 32 college students at the University of Oklahoma (Mage = 19.4318 years) who participated in one 60-min session in partial fulfillment of class requirements for an intro-319 ductory psychology course. Results from these students served as a benchmark against which data320 from 91 newly tested children in the current experiments were compared. There were four groups321 of children who each participated in two 30-min sessions. See Table 1 for participant demographics.322 In total, 19 first graders, 5 second/third graders, and 4 fourth/fifth graders were excluded from anal-323 yses because (a) they made a substantial proportion of fast responses (6400 ms) showing that they324 were just hitting keys as fast as possible and had given up on the task or (b) they did not complete325 enough trials for modeling. Not completing enough trials for modeling was particularly a problem326 for first graders. These children either lacked an ability to complete the task or found the tasks to327 be very difficult/monotonous and, thus, decided to give up.

328 Procedure and stimuli

329 Participants completed the symbolic number discrimination task during Session 1 and completed330 the non-symbolic numerosity discrimination task during Session 2. Stimuli were displayed on the

Table 1Participant demographics.

Grade n Mean age (SD) Sex Ethnicity breakdown Mean days betweentesting sessions

First 19 7.12 (0.21) 53% girls 79% Caucasian, 16% biracial, 5% Asian 10.01Second/Third 26 8.20 (1.83) 62% girls 85% Caucasian, 8% Asian, 4% biracial, 4% Hispanic 22.62Fourth/Fifth 27 10.46 (0.63) 59% girls 89% Caucasian, 4% African American, 4% Native

American, 4% biracial28.59

Seventh/Eighth 19 13.22 (0.58) 37% girls 63% Caucasian, 11% Native American, 11% biracial,5% African American, 5% Asian, 5% Hispanic

2.47



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331 screen of a laptop computer, and responses were collected from the laptop’s keyboard using the ‘‘?”332 key for ‘‘large” responses and the ‘‘Z” key for ‘‘small” responses. Prior to beginning each experiment,333 participants were told that sometimes decisions would be difficult to make, but they were to respond334 as quickly and accurately as possible. Participants were encouraged to take a brief rest break between335 each block of trials. During the rest break, participants saw two progress bars that displayed their pro-336 portion of correct and error responses.337 During Session 3, second/third and fourth/fifth graders completed the matrix reasoning and vocab-338 ulary subtests of the Wechsler Intelligence Scale for Children–Fourth Edition (WISC-IV). The second/339 third graders from our sample had a mean IQ score of 115 on the matrix reasoning and vocabulary340 subtests; the fourth/fifth graders had a mean IQ score of 117. We do not have IQ scores from the first341 graders, the seventh/eighth graders, or the college-aged adults. However, the IQ scores of the second/342 third and fourth/fifth graders are almost the same as those of a comparable college-aged sample343 (M = 119) who completed the Wechsler Adult Intelligence Scale–Revised (WAIS-R) matrix reasoning344 and vocabulary subtests (Ratcliff et al., 2001). Average achievement test scores (Basic Early Assess-345 ment of Reading [BEAR] and Oklahoma Core Curriculum Tests [OCCT]), obtained from the public346 school district administrators, and average IQ scores are listed in the Appendix.

347 Symbolic number discrimination348 White Arabic numerals between 10 and 90 were displayed on a black background in the middle of a349 laptop screen. Participants pushed the ‘‘Z” key if a small number between 10 and 49 was displayed and350 pushed the ‘‘?” key if a large number between 51 and 90 was displayed. Each numeral remained vis-351 ible until a response key was pressed. After a key was pressed, the screen was cleared, a smiling (cor-352 rect response) or frowning (incorrect response) face was displayed for 500 ms, a blank screen353 appeared for 100 ms, and then the next trial began. There were eight blocks of trials, 80 trials per354 block, with each of the possible numbers tested once in each block in random order.

355 Non-symbolic numerosity discrimination356 On each trial, some number of white asterisks, between 11 and 90, was displayed against a black357 background. The asterisks occupied randomly selected positions in a 10 � 10 grid in the center of the358 laptop screen that subtended a visual angle of 7.5 degrees horizontally and 7.0 degrees vertically.359 Arrays containing fewer than 11 asterisks were not included to deter participants from counting.360 Participants pushed the ‘‘Z” key if a small number of asterisks between 11 and 49 was displayed361 and pushed the ‘‘?” key if a large number of asterisks between 51 and 90 was displayed. The asterisks362 remained on the screen until a response key was pressed. Then the screen was cleared, a smiling (cor-363 rect response) or frowning (incorrect response) face was displayed for 500 ms, the screen was cleared,364 a blank screen appeared for 100 ms, and then the next trial began. There were eight blocks of trials, 80365 trials per block, with each of the possible numbers of asterisks tested once in each block in random366 order.

367 Results

368 Responses shorter than 300 ms for children and 250 ms for adults were eliminated from analyses,369 as were responses longer than 4000 ms for children and 2000 ms for adults. The percentages elimi-370 nated were 19.1%, 6.1%, 5.1%, 4.4%, and 3.0% for first graders, second/third graders, fourth/fifth graders,371 seventh/eighth graders, and college-aged adults, respectively, for the numerosity discrimination task,372 and 11.5%, 3.4%, 1.6%, 1.1%, and 3.0% for these corresponding groups in the number discrimination373 task.374 Table 2 shows the proportion correct and mean RT for children in first, second/third, fourth/fifth,375 and seventh/eighth grades and college-aged adults in the numerosity and number discrimination376 tasks. The left-most column of Table 2 shows that experimental stimuli (e.g., number of asterisks or377 numerals presented on the computer screen) were grouped into conditions (six for the numerosity378 discrimination task and eight for the number discrimination task) to provide more observations per379 condition for application of the diffusion model. The conditions were created by grouping ‘‘small”



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380 responses to small stimuli with ‘‘large” responses to large stimuli (e.g., 17–22 asterisks grouped with381 71–76 asterisks). In this way, our conditions collapsed over absolute distance from the standard (e.g.,382 50 asterisks). In addition, we made sure that ‘‘small” responses to small stimuli were similar to ‘‘large”383 responses to large stimuli. Recent computational modeling results suggest that an interaction of stim-384 ulus ratio (e.g., Weber fraction) and absolute distance best explains adults’ performance on non-385 symbolic numerical discrimination tasks (Prather, 2014). But this appears to be task specific. In our386 numerosity and number discrimination tasks, ‘‘large” responses to large stimuli are symmetric to387 ‘‘small” responses to small stimuli (Ratcliff, 2014). Thus, using stimulus ratios would combine condi-388 tions with data that are not equivalent.389 The participants were slightly biased in the numerosity discrimination task such that their chance390 level of accuracy for responding ‘‘small” and ‘‘large” was approximately 47 and not 50 (as in Ratcliff,391 2014, Experiments 1 and 2; Ratcliff et al., 2015). It is for this reason that there are slightly more num-392 bers in the first bin as compared with subsequent bins shown in Table 2.393 Note that we refer interchangeably to age and grade because older children were in higher grades394 than younger children. However, we did not use age as a continuous variable in any of the analyses.395 Across all grades on both the numerosity and number discrimination tasks, accuracy increased and RT396 decreased as difficulty level decreased. That is, the closer the stimuli were to 50, the lower the397 accuracy and the longer the RT. Participant accuracy was at or near ceiling levels for the three easiest398 conditions in the numerosity discrimination task that contained the most/least asterisks (.799–.894399 for first graders, .876–.936 for second/third graders, .891–.938 for fourth/fifth graders, .904–.947 for sev-400 enth/eighth graders, and .908–.943 for adults). RTs continued to decrease in the three easiest conditions401 as the stimuli becamemore discriminable, that is, as their distance from 50 increased (1076–994 ms for402 first graders, 911–845 ms for second/third graders, 795–705 ms for fourth/fifth graders, 660–593 ms for403 seventh/eighth graders, and 491–455 ms for adults). In the number discrimination task, accuracy was404 near ceiling, 80% to 97% range, for each of the conditions (.802–.906 for first graders, .854–.955 for405 second/third graders, .858–.957 for fourth/fifth graders, .870–.970 for seventh/eighth graders, and406 .879–.972 for adults). As in thenumerositydiscrimination task, RTs fromthenumberdiscrimination task407 decreased substantially as the conditions went from hardest to easiest (1302–1135 ms for first408 graders, 1239–1015 ms for second/third graders, 1034–811 ms for fourth/fifth graders, 758–629 ms409 for seventh/eighth graders, and 632–501 ms for adults).410 Fig. 2 plots accuracy and median RT against level of difficulty. Note that the most difficult411 conditions were the ones with the smallest absolute distance from 50. For both tasks, accuracy412 decreased as a function of difficulty, although the decrease in accuracy was more apparent for the

Table 2Response proportions and mean RTs for the numerosity and number discrimination tasks.

Task Condition 1 grade 2/3 grade 4/5 grade 7/8 grade Adult

Pr.Cor.

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Mean Cor.RT

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Mean Cor.RT

Pr.Cor.

Mean Cor.RT

Numerosity 11–16/77–88 .894 994 .936 845 .938 705 .947 593 .943 45517–22/71–76 .856 1049 .916 895 .909 737 .940 623 .934 47923–28/65–70 .799 1076 .876 911 .891 795 .904 660 .908 49129–34/59–64 .757 1110 .813 959 .825 838 .846 701 .874 52735–40/53–58 .671 1109 .710 1009 .735 864 .753 747 .783 55841–46/47–52 .570 1139 .590 1034 .594 901 .600 799 .617 591

Number 10–14/86–90 .906 1135 .955 1015 .957 811 .970 629 .972 50115–19/81–85 .887 1179 .930 1016 .950 861 .948 616 .963 51220–24/76–80 .867 1155 .934 1029 .945 868 .956 616 .968 51025–29/71–75 .852 1203 .938 1056 .943 867 .948 637 .952 52930–34/66–70 .860 1209 .916 1099 .929 904 .937 664 .956 53435–39/61–65 .854 1246 .908 1149 .919 946 .924 676 .943 56340–44/56–60 .821 1253 .859 1176 .880 1000 .896 723 .904 59045–49/51–55 .802 1302 .854 1239 .858 1034 .870 758 .879 632

Note. Pr. Cor., proportion correct; Mean Cor. RT, mean correct response time.



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413 non-symbolic task than for the symbolic task in which performance was at ceiling. RT also increased as414 difficulty increased across both tasks. There was no evidence of a size effect, with smaller numbers415 discriminated from the 50 cutoff better than larger numbers (also see Figs. 2 and 4 in Ratcliff,416 2014). Dehaene, Dupoux, and Mehler (1990) provided evidence that 1 and 2 are easier to discriminate417 from one another than are larger numbers equidistant apart, such as 8 and 9, but we did not see this418 result with the larger numbers used in our experiments. This may also reflect a task difference; judg-419 ing differences between two numbers or numerosities is different from judging whether one number420 or numerosity is different from a criterion number or numerosity. Importantly, our numerosity task is421 tapping a basic numeracy skill because it has been shown to correlate with number discrimination as422 well as number memory tasks (Ratcliff et al., 2015).423 When accuracy was averaged across all of the easy and difficult conditions, first graders were less424 accurate than all other groups on the numerosity and number discrimination tasks (see Table 3). There425 were no other significant differences in averaged accuracy across the groups of participants. These426 data averaged across conditions indicated that children in second/third grade were responding as427 accurately as older children and adults on the two tasks; however, there were significant grade428 differences in median RT averaged across conditions (see Table 3) for both tasks. First through third

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Fig. 2. The top panels are plots of average accuracy across conditions for the numerosity (left) and number (right)discrimination tasks for first, second/third, fourth/fifth, and seventh/eighth graders and adults. The bottom panels are plots ofmedian RT across conditions for the numerosity (left) and number (right) discrimination tasks. As task difficulty increased (e.g.,arrays of dots/numbers near 50), participants’ accuracy decreased and their RTs increased.



10 May 2016



429 graders did not differ from one another in median RTs for the number discrimination task, nor did sev-430 enth/eighth graders and adults. Fourth/fifth graders, seventh/eighth graders, and adults made faster431 responses than did first graders and second/third graders. Seventh/eighth graders and adults made432 faster responses than did fourth/fifth graders. For the numerosity discrimination task, participants433 in all grades made faster responses than did first graders. Fourth/fifth graders, seventh/eighth graders,434 and adults made faster responses than did second/third graders. Seventh/eighth graders and adults435 made faster responses than did fourth/fifth graders. Adults made faster responses than did seventh/436 eighth graders.

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Fig. 3. Model predictions plotted against experimental data for the numerosity discrimination task. Accuracy and RT data from.1, .5, and .9 quantiles are plotted.



10 May 2016



437 The model was fit to the data for each participant individually using a standard chi-square method.438 The values of all the components of processing identified by the model are estimated simultaneously439 from the data for all the conditions in an experiment. The fittingmethod uses quantiles of the RT distri-440 butions for correct and error responses for each condition (the .1, .3, .5, .7, and .9 quantile RTs). Themodel441 predicts the cumulative probability of a response at each RT quantile. Subtracting the cumulative442 probabilities for each successive quantile from the next higher quantile gives the proportion of443 responses between adjacent quantiles. For a chi-square computation, these are the expected values,444 to be compared with the observed proportions of responses between the quantiles (i.e., the

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1 gr

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Fig. 4. Model predictions plotted against experimental data for the number discrimination task. Accuracy and RT data from .1,.5, and .9 quantiles are plotted.



10 May 2016



445 proportionsbetween .1, .3, .5, .7, and .9 are each .2, and theproportionsbelow .1 andabove .9 are both .1).446 Summingover (Observed � Expected)2/Expected for correct anderror responsesmultipliedby thenum-447 ber of observations for each condition gives a single chi-square value, and the sum of these over condi-448 tions is minimized with a general SIMPLEX minimization routine. The parameter values for the model449 are adjusted by SIMPLEX until the minimum chi-square value is obtained (see Ratcliff & Tuerlinckx,450 2002, and Ratcliff & Childers, 2015, for a full description of the fitting method).451 The RTs used for fitting the model, especially for the number discrimination task, were slightly452 complicated by the fact that for many adult participants, for many conditions, there were fewer than453 five errors and so quantiles could not be computed. When this was the case, the RT distribution for a454 condition was divided at its median, and the model was fit by predicting the cumulative probability of455 responses above and below the median. This reduced the number of degrees of freedom from 6 to 2 for456 that error condition. (To avoid very small or very large medians when there were only one or two457 responses, when these might be outliers, this division was used only when the median for errors458 was between the .3 and .7 median RTs for correct responses.) This reduction procedure was not used459 on the child data because these participants committed enough errors to allow their data to be mod-460 eled in the typical way. Mean chi-square values (Table 4) were lower than critical values. Because of461 the conditions with fewer than five errors, the average degrees of freedomwere reduced from 55 to 50462 for the numerosity discrimination task and from 74 to 54 for the number discrimination task for the463 adult participants.

Table 3Grade differences in means across conditions for average accuracy (proportion correct) and median RT (ms) in the numerosity andnumber Tasks.

Dependent variable F statistic Post-hoc t-tests (Bonferroni) Post-hoc p-values

Accuracy for numerositytask

F(4,118) = 7.87, p < .0001,g2 = .211

First (.77) < Second/Third (.81) p < .05First (.77) < Fourth/Fifth (.82) p < .01First (.77) < Seventh/Eighth (.82) p < .05First (.77) < Adults (.85) p < .0001

Accuracy for number task F(4,118) = 8.27, p < .0001,g2 = .219

First (.86) < Second/Third (.92) p < .01First (.86) < Fourth/Fifth (.93) p < .0001First (.86) < Seventh/Eighth (.93) p < .01First (.86) < Adults (.94) p < .0001

RT for numerosity task F(4,118) = 68.14, p < .0001,g2 = .698

First (950.62) > Second/Third (819.11) p < .01First (950.62) > Fourth/Fifth (692.50) p < .0001First (950.62) > Seventh/Eighth (585.56) p < .0001First (950.62) > Adults (473.48) p < .0001Second/Third (819.11) > Fourth/Fifth(692.50)

p < .01

Second/Third (819.11) > Seventh/Eighth(585.56)

p < .0001

Second/Third (819.11) > Adults (473.48) p < .0001Fourth/Fifth (692.50) > Seventh/Eighth(585.56)

p < .05

Fourth/Fifth (692.50) > Adults (473.48) p < .0001Seventh/Eighth (585.56) > Adults (473.48) p < .01

RT for number task F(4,118) = 69.71, p < .0001,g2 = .70

First (1081.83) > Fourth/Fifth (783.04) p < .0001First (1081.83) > Seventh/Eighth (590.98) p < .0001First (1081.83) > Adults (494.51) p < .0001Second/Third (964.28) > Fourth/Fifth(783.04)

p < .0001

Second/Third (964.28) > Seventh/Eighth(590.98)

p < .0001

Second/Third (964.28) > Adults (494.51) p < .0001Fourth/Fifth (783.04) > Seventh/Eighth(590.98)

p < .0001

Fourth/Fifth (783.04) > Adults (494.51) p < .0001



10 May 2016



464 Figs. 3 and 4 demonstrate that the model accounted for the data in both the numerosity and num-465 ber discrimination tasks well. These figures plot the predicted values from the diffusion model against466 the experimental values for accuracy at the .1, .5, and .9 quantile RTs. A developmental progression is467 clear for both tasks. The predicted values from the model are a closer fit for older children and adults468 than for the very youngest participants.

469 Model-based interpretations of the data

470 The values of the parameters that generated the best fit of the model to the data averaged over par-471 ticipants are shown in Tables 4 and 5. It should be noted that the mean parameter values across the472 two tasks were fairly similar. For example, the boundaries that first graders set on the numerosity dis-473 crimination task were similar to the boundary separations that they set on the number discrimination474 task. Likewise, non-decision times were similar across tasks, as were drift rates for equivalent475 conditions (e.g., the easiest conditions in which numbers or arrays of asterisks were furthest from476 50). Similar parameter settings across the two tasks for individuals at each grade level indicate that477 the tasks are likely tapping a similar underlying numerical ability (see Ratcliff et al., 2015). Statistical478 analyses comparing the parameter values across grade levels in the numerosity and number discrim-479 ination tasks are included in Tables 6 and 7, respectively. First we discuss the parameter values aver-480 aged over participants, and then we discuss individual differences in the parameter values.

481 Values of model parameters averaged across participants

482 Fig. 5 shows the psychometric functions that relate drift rate (v) to difficulty for the numerosity483 (top) and number (bottom) discrimination tasks. Average drift rate (y-axis) is plotted against the inde-484 pendent variable of number of asterisks or number shown on the computer screen (x-axis). Linear485 functions characterize each grade level for both experiments. Performance is near chance for numbers486 of asterisks around 50, so the intercept of the drift rate function is near zero for the numerosity dis-487 crimination task. The intercept is well above zero for the number discrimination task. Children in488 lower grades had significantly lower drift rates than did children in higher grades and adults on both489 the numerosity and number discrimination tasks. This indicates that numerical acuity/discrimination490 abilities increased with increasing grade level. Children in lower grades set wider boundaries (a) than491 did children in higher grades and adults for the numerosity and number discrimination tasks (see492 Fig. 6, top left panel). Thus, the decision criteria narrowed with increasing grade level. Children in493 lower grades had larger non-decision times (Ter) than did children in higher grades and adults for both

Table 4Diffusion model parameters.

Task Group a Ter g sz st v2

Numerosity 1 grade .195 .450 .074 .102 .293 69.62/3 grade .193 .429 .125 .092 .249 64.64/5 grade .170 .396 .131 .085 .248 67.17/8 grade .154 .380 .180 .085 .134 56.2Adult .117 .322 .134 .067 .156 64.3

Number 1 grade .210 .497 .034 .109 .290 99.92/3 grade .227 .462 .085 .102 .209 84.84/5 grade .194 .420 .106 .085 .180 75.77/8 grade .152 .392 .124 .077 .150 73.1Adult .129 .336 .099 .072 .140 64.2

Note. The parameters were as follows: boundary separation a (starting point z = a/2), mean non-decision component of responsetime Ter, SD in drift across trials g, range of the distribution of starting point sz, and range of the distribution of non-decisiontimes st. Critical values of chi-squares are 67.5 for 50 degrees of freedom for the numerosity discrimination task and 72.2 for54 degrees of freedom for the number discrimination task.



10 May 2016



494 the numerosity and number discrimination tasks (see Fig. 6, bottom left panel). That is, with increas-495 ing grade, participants were quicker at encoding, transforming the stimulus array, and executing their496 decisions. There was a significant grade difference in the range in non-decision times (st) across trials497 for both the numerosity and number discrimination tasks (see Fig. 6, bottom left panel). Variability in498 non-decision time got smaller as grade level increased. There was a significant grade difference in the499 standard deviation in drift across trials (g) for the numerosity and number discrimination tasks (see500 Fig. 6, top right panel). This means that the numerical acuity/discrimination became less variable with501 increasing grade level. There was no statistically significant difference in the range in across-trial vari-502 ability in starting point (sz) across grade levels for the numerosity task; however, there was a signif-503 icant difference in sz across grade levels for the number discrimination task, with children in lower504 grades showing greater variability in starting points across trials than adults (see Fig. 6, bottom right505 panel).

506 Differences among individuals in data and model parameters

507 The more accurate the participant was on the numerosity discrimination task, the more accurate508 the participant was on the number discrimination task (see Table 8), and this was true across grade509 levels. All of these correlations were significant at the p < .05 level except for the performance of first510 graders. Similarly, the faster a participant responded (mean RT) on the numerosity discrimination511 task, the faster the participant responded (mean RT) on the number discrimination task (see Table 8).512 Again, this was true across grades. As reported previously by Ratcliff and colleagues (2015), there were513 extremely weak correlations between mean RT and accuracy for the numerosity discrimination task514 for all grades except seventh/eighth in which a moderate, but not statistically significant, correlation515 was found. Similarly, there were weak correlations betweenmean RT and accuracy on the number dis-516 crimination task for all grades except fourth/fifth in which a moderate statistically significant corre-517 lation was found. The most accurate participants were not always the participants who responded518 the quickest. Importantly, the common belief would be that accuracy and RT are negatively correlated519 with one another (e.g., accurate participants produce shorter response times). The correlations that520 were found between accuracy and RT, however, were positive instead of negative. In fact, only 2 of521 the 10 correlations comparing accuracy and RT on the numerosity and number discriminations tasks522 were negative (see Table 8).523 Fig. 7 shows scatter plots and the correlations between model parameters (e.g., drift rate on the524 numerosity discrimination task vs. drift rate on the number discrimination task) across the two tasks525 for each grade. Drift rate was highly correlated across the two tasks for all grade levels. Non-decision526 time was highly correlated across tasks for first graders and seventh/eighth graders and was moder-527 ately so for second/third graders and fourth/fifth graders. Boundary separation was strongly correlated528 across tasks for first graders and fourth/fifth graders and was moderately correlated for second/third529 graders and seventh/eighth graders. These results showed that the main components of processing

Table 5Drift rates.

Task Group v1 v2 v3 v4 v5 v6 v7 v8

Numerosity 1 grade .193 .163 .123 .095 .062 .0222/3 grade .268 .228 .194 .145 .092 .0424/5 grade .314 .265 .218 .161 .110 .0357/8 grade .414 .382 .298 .222 .144 .059Adult .482 .392 .351 .249 .167 .060

Number 1 grade .142 .127 .116 .100 .100 .102 .079 .0722/3 grade .218 .210 .203 .196 .166 .151 .118 .1084/5 grade .304 .272 .248 .255 .227 .194 .147 .1247/8 grade .391 .361 .370 .341 .285 .290 .230 .189Adult .434 .409 .414 .365 .346 .305 .253 .189



10 May 2016



530 were related across tasks. If an individual had a high drift rate, large boundary separation, or long non-531 decision time on one task, the individual tended to have a high drift rate, large boundary separation, or532 long non-decision time on the other task.533 The diffusion model parameters were not highly correlated with one another, and this was true534 across grade level (see Table 9). That is, boundary settings were not strongly correlated with non-535 decision times or drift rates, and non-decision times were not strongly correlated with drift rates.536 The only exception was that for first graders non-decision time and drift rate were significantly cor-537 related. These results are consistent with those reported by Ratcliff and colleagues (2015), in which it538 was suggested that the diffusion model decomposes the dependent variables—accuracy and correct539 and error RT distributions—into separate and relatively independent components of processing.

Table 6Grade differences in parameter values for numerosity task.

Parameter F statistic Post-hoc t-tests (Bonferroni) Post-hoc p-values

Drift rate (v) F(4,118) = 19.27, p < .0001,g2 = .395

First (.093) < Fourth/Fifth (.158) p < .05First (.093) < Seventh/Eighth(.221)

p < .0001

First (.093) < Adults (.247) p < .0001Second/Third (.14) < Seventh/Eighth (.221)

p < .01

Second/Third (.14) < Adults (.247) p < .0001Fourth/Fifth (.158) < Seventh/Eighth (.221)

p < .05

Fourth/Fifth (.158) < Adults (.247) p < .0001

Boundaries (a) F(4,118) = 14.83, p < .0001,g2 = .334

First (.20) > Seventh/Eighth (.154) p < .05First (.20) > Adults (.118) p < .0001Second/Third (.193) > Seventh/Eighth (.154)

p < .05

Second/Third (.193) > Adults(.118)

p < .0001

Fourth/Fifth (.17) > Adults (.118) p < .0001Seventh/Eighth (.154) > Adults(.118)

p = .051

Non-decision times (Ter) F(4,118) = 29.96, p < .0001,g2 = .504

First (.45) > Fourth/Fifth (.396) p < .01First (.45) > Seventh/Eighth (.38) p < .0001First (.45) > Adults (.323) p < .0001Second/Third (.429) > Seventh/Eighth (.38)

p < .01


p < .0001

Fourth/Fifth (.396) > Adults (.323) p < .0001Seventh/Eighth (.38) > Adults(.323)

p < .0001

Range in non-decision times acrosstrials (st)

F(4,118) = 19.63, p < .0001,g2 = .40

First (.293) > Seventh/Eighth(.134)

p < .0001

First (.293) > Adults (.154) p < .0001Second/Third (.249) > Seventh/Eighth (.134)

p < .0001


p < .0001

Fourth/Fifth (.248) > Seventh/Eighth (.134)

p < .0001

Fourth/Fifth (.248) > Adults (.154) p < .0001

Standard deviation in drift ratesacross trials (g)

F(4,118) = 3.16, p < .05,g2 = .096

First (.074) < Seventh/Eighth (.18) p < .01

Range of starting points across trials(sz)

F(4,118) = 1.86, p > .05



10 May 2016



540 Table 10 shows correlations among the behavioral measures, RT and accuracy, and model param-541 eters. Our previous work with adults (Ratcliff et al., 2015) showed that accuracy is mainly determined542 by drift rate and RT is mainly determined by boundary settings, and that was replicated across grade543 levels in our current experiments.

544 Discussion

545 Previous research (Ratcliff et al., 2012) showed a developmental progression in children’s simple546 two-choice non-symbolic numerical discrimination decisions. For example, children accumulated a547 lower quality of evidence than did adults when attempting to make a non-symbolic numerical deci-548 sion, set wider decision criteria than did adults, and were slower at encoding the stimulus array, trans-549 forming the array into task-relevant information, and executing a motor response than were adults.550 One goal of our current experiments was to determine whether this pattern of results could be repli-551 cated with a new group of children and extended to a symbolic number discrimination task. We also552 investigated whether RT and accuracy on the symbolic and non-symbolic discrimination tasks was553 related across individuals. Indeed, our current results provide evidence for a developmental progres-554 sion in diffusion model parameters toward higher drift rates, narrower boundary settings, and faster555 non-decision components across the lifespan in both non-symbolic and symbolic tasks. In contrast,556 previous research found that only boundary separation and non-decision criteria differed between

Table 7Grade differences in parameter values for number task.

Parameter F statistic Post-hoc t-tests (Bonferroni) Post-hoc p-values

Drift rate (v) F(4,118) = 37.77,p < .0001, g2 = .561

First (.10) < Second/Third (.165) p < .05First (.10) < Fourth/Fifth (.21) p < .0001First (.10) < Seventh/Eighth (.295) p < .0001First (.10) < Adults (.326) p < .0001Second/Third (.165) < Seventh/Eighth (.295) p < .0001Second/Third (.165) < Adults (.326) p < .0001Fourth/Fifth (.21) < Seventh/Eighth (.295) p < .01Fourth/Fifth (.21) < Adults (.326) p < .0001

Boundaries (a) F(4,118) = 19.8,p < .0001, g2 = .402

First (.211) > Seventh/Eighth (.152) p < .01First (.211) > Adults (.129) p < .0001Second/Third (.227) > Seventh/Eighth (.152) p < .0001Second/Third (.227) > Adults (.129) p < .0001Fourth/Fifth (.194) > Seventh/Eighth (.152) p < .05Fourth/Fifth (.194) > Adults (.129) p < .0001

Non-decision times (Ter) F(4,118) = 33.6,p < .0001, g2 = .532

First (.497) > Fourth/Fifth (.42) p < .0001First (.497) > Seventh/Eighth (.392) p < .0001First (.497) > Adults (.336) p < .0001Second/Third (.462) > Seventh/Eighth (.392) p < .0001Second/Third (.462) > Adults (.336) p < .0001Fourth/Fifth (.42) > Adults (.336) p < .0001Seventh/Eighth (.392) > Adults (.336) p < .01

Range in non-decisiontimes across trials (st)

F(4,118) = 9.73,p < .0001, g2 = .248

First (.29) > Second/Third (.209) p < .05First (.29) > Fourth/Fifth (.18) p < .001First (.29) > Seventh/Eighth (.15) p < .0001First (.29) > Adults (.14) p < .0001Second/Third (.209) > Adults (.14) p < .05

Standard deviation in driftrates across trials (g)

F(4,118) = 4.05,p < .01, g2 = .121

First (.034) < Fourth/Fifth (.106) p < .05First (.034) < Seventh/Eighth (.124) p < .01First (.034) < Adults (.099) p < .05

Range of starting pointsacross trials (sz)

F(4,118) = 4.02,p < .01, g2 = .119

First (.109) > Adults (.072) p < .05Second/Third (.102) > Adults (.072) p < .05



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557 aging adults and college-aged students who completed a non-symbolic discrimination task (Ratcliff558 et al., 2001). Our current findings indicate that with increasing age, children’s symbolic and non-559 symbolic magnitude knowledge improves and results in higher drift rates.560 RTs decreased and accuracy increased from the hardest conditions (comparisons closest to 50) to561 the simplest conditions (comparisons furthest from 50), and this provided evidence for the numerical562 distance effect (Dehaene et al., 1998) in both the symbolic and non-symbolic discrimination tasks.563 Across age ranges, participants were at ceiling levels of accuracy on the number discrimination task564 and also on the simplest conditions in the numerosity discrimination task. By the time children are565 in first grade, they have already acquired some adult-like abilities in comparing Arabic numerals566 and asterisk arrays.567 We found (in line with Ratcliff et al., 2015) that RT was strongly correlated across the symbolic568 number discrimination task and the non-symbolic numerosity discrimination task, as was accuracy.569 This means that participants who made fast responses on one task tended to make fast responses570 on the other task, and participants who made accurate decisions on one task made accurate decisions571 on the other task. Diffusion model drift rate, boundary separation, and non-decision time parameters572 were also strongly correlated with the same model parameter on the two tasks. For example, a

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573 participant who had a high drift rate (acuity) on one task was able to abstract high-quality information574 from the stimulus array on the other task, and a participant with conservative boundary settings on575 one task tended to set conservative decision criteria on the other task.576 In contrast, RT and accuracy were not correlated with each other within a task so that whether a577 participant was fast did not predict whether the participant was accurate. In addition, drift rate,578 boundary setting, and non-decision time parameters were not correlated with each other within a

Age (years) Age (years)

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Fig. 6. Developmental differences in boundary separation (upper left panel), SD in drift rate (upper right panel), non-decisiontime and across-trial range in non-decision time (bottom left panel), and across-trial range in starting point (bottom rightpanel).

Table 8Correlations for accuracy and mean RT averaged across conditions.

Group Accuracy–accuracy Mean RT–mean RT Numerosity accuracy–mean RT Number accuracy–mean RT

1 grade .372 .667 �.049 .3352/3 grade .609 .822 .045 �.2364/5 grade .341 .838 .187 .4807/8 grade .762 .550 .404 .299Adult .634 .592 .078 .205

Note. Critical values for the correlations are as follows: first grade, .46; second/third grade, .39; fourth/fifth grade, .38; seventh/eighth grade, .46; and adult, .35.



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579 task. This provided an explanation within the diffusion model framework of the lack of correlation580 between RT and accuracy (see Ratcliff et al., 2015). That is, the model successfully differentiates the581 quality of information on which participants base their decisions from their willingness to make deci-582 sions and other components of information processing such as their ability to encode and transform a583 stimulus and execute a decision.584 Our results may help to reconcile discrepant findings in the numerical cognition literature. Some-585 times there are significant correlations across symbolic and non-symbolic tasks in the numerical cog-586 nition literature, and sometimes there are not. Fazio and colleagues’ (2014) meta-analysis indicated

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Fig. 7. Plots of model parameters between the numerosity and number discrimination tasks for boundary separation (leftcolumn), non-decision time (middle column), and drift rate (right column). The critical values for the correlations are as follows:first graders = .46, second/third graders = .39, fourth/fifth graders = .38, seventh/eighth graders = .46, and adults = .35.



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587 the circumstances under which correlations between non-symbolic numerical discrimination tasks588 and mathematics achievement are likely to be found. There were stronger correlations between589 non-symbolic task performance and mathematics achievement when accuracy-based measures or590 accuracy- and RT-based measures of non-symbolic task performance were combined rather than591 when RT-based measures of non-symbolic task performance were considered alone.592 Therefore, the discrepancy in the numerical cognition literature may at least partly stem from the593 fact that researchers primarily use either accuracy- or RT-based measures, but rarely both, to describe594 performance on numerical decision tasks. The classic study of symbolic numerical discrimination con-595 ducted by Sekuler and Mierkiewicz (1977) is a prime example in which changes in RT only were596 tracked across age cohorts. Our results point to the fact that accuracy and RT vary independently597 and may stem from different underlying processes; we found no evidence in support of the common598 belief that accurate participants should make faster responses relative to the average values for the599 group. Therefore, basing claims about participants’ numerical decision making on either of these600 dependent variables alone may lead to a skewed understanding of children’s and adults’ numerical601 information processing abilities. The explanation for the lack of negative correlations between RT602 and accuracy within the framework of the diffusion model was that drift rates, boundary settings,603 and non-decision times are independent of one another, accuracy is mainly determined by drift rate,604 and RT is mainly determined by boundary settings. One advantage of modeling the data with the dif-605 fusion model is that it accounts for the quality of the accumulated information (drift rate) separately606 from speed/accuracy settings (boundary separation).607 Ceiling- and floor-level performance may also contribute to the inconsistent findings in the numer-608 ical cognition literature. That is, if all participants make highly accurate decisions but there is some609 variability in RTs, researchers may then choose to report RTs instead of accuracy because of the610 increased variability in the RT data as compared with the accuracy data. Drift rates can be obtained611 for participants when their RTs vary even though their accuracy may be at ceiling or floor levels.612 Our current results indicate that an information-processing model, such as the diffusion model,613 that decomposes simple two-choice decisions into an encoding/response execution component, an614 evidence accumulation component, and speed/accuracy settings can inform theories about numerical615 decision making across the lifespan. Our model-based analysis of children’s performance on symbolic616 and non-symbolic numerical decisions suggests evidence that the number and numerosity discrimi-617 nation tasks are tapping participants’ underlying numerical ability. Future research should investigate618 whether diffusion model parameters are correlated with mathematics achievement test scores and IQ619 scores. These analyses could point to the mechanisms by which basic numeracy skills, such as decid-620 ing whether an Arabic numeral is greater than or less than 50, are related to more complex mathemat-621 ical abilities indexed on achievement tests. We were unable to obtain achievement test scores for all of622 our participants because the very youngest children were not state-mandated to take achievement623 tests, and not all parents gave permission for their children’s scores to be released for research pur-624 poses. Given findings from Chen and Li’s (2014) meta-analysis, we were concerned that any significant

Table 9Correlations between model parametersaveraged across the two tasks.

Group a/Ter a/v Ter/v

1 grade �.40 �.24 .512/3 grade �.23 .21 �.084/5 grade �.27 .09 .097/8 grade �.17 .27 .19Adult .16 �.13 .18

Note. Critical values for the correlations areas follows: first grade, .46; second/thirdgrade, .39; fourth/fifth grade, .38; seventh/eighth grade, .46; and adult, .35.



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625 or non-significant correlations that we found between non-symbolic task performance and mathe-626 matics achievement would be underpowered in our sample. One ‘‘proof of existence” example from627 the domain of reading illustrates that in a large individual differences study with struggling adult628 readers, drift rates from a lexical decision task were correlated with standardized reading scores629 and IQ scores (McKoon & Ratcliff, 2016). Future research should involve the testing of many more chil-630 dren from each grade level to establish whether there are also significant relations among model631 parameters for symbolic and non-symbolic numerical decision tasks and standardized achievement/632 IQ scores in the domain of mathematics. This could be especially important to tease apart the reasons633 why so many first graders were unable to complete the tasks (e.g., general lack of ability vs. task dif-634 ficulty vs. boredom).635 It is an open question as to whether diffusion model parameters that model non-symbolic discrim-636 ination data are related to overall mathematics achievement. Like Fazio and colleagues (2014), we637 hypothesize that simple two-choice symbolic and non-symbolic numerical decision tasks may tap638 more rudimentary information processing skills than the more complex variety of skills necessary639 to answer questions on standardized achievement tests and IQ assessments. Standardized tests of640 mathematics achievement may test a range of skills, some of which are related to basic numerical641 abilities and some of which are unrelated to numerical abilities. Future research could focus on642 cataloging the constructs being tested on standardized achievement tests such that the same basic643 underlying processes at play in the achievement tests can be more directly compared with similar644 cognitive tasks.645 Another step for future research will be to use stimuli that control for continuous extent (e.g., yel-646 low and blue dot arrays available from http://panamath.org) to determine whether diffusion model647 parameters differ when these aspects of the stimuli are controlled. Pilot data that we have collected648 in our labs with college-aged adults have shown that drift rates for non-symbolic tasks that do and649 do not control for continuous extent are correlated. In addition, drift rates on the symbolic and650 non-symbolic tasks used in the current experiments are highly correlated for participants across all651 of the grades we tested; therefore, these tasks are likely tapping a common underlying numerical652 ability.

653 Uncited reference

654 Halberda and Feigenson (2008).

655 Acknowledgments

656 The authors thank Brian Pickens, Mary Rischard, Sarah Wiatrek, and Heather Smith for their help657 with data collection and thank Ashley Stafford and Craig Cruzan for their help in administering the658 WISC-IV subtests.

Table 10Correlations of model parameters and accuracy and RT across grades.

Accuracy for numerositydiscrimination

Accuracy for numberdiscrimination

RT for numerositydiscrimination

RT for numberdiscrimination

a Ter V a Ter V a Ter V a Ter V

1 grade �0.013 0.464 0.613 0.165 0.414 0.814 0.843 �0.316 �0.581 0.632 0.095 �0.2052/3 grade 0.055 �0.067 0.419 0.116 0.299 0.542 0.749 0.172 �0.421 0.227 0.331 �0.6524/5 grade 0.51 �0.237 0.474 0.554 �0.25 �0.068 0.536 �0.027 �0.367 0.882 �0.048 �0.6377/8 grade 0.48 �0.154 0.56 0.46 0.017 0.456 0.79 �0.058 �0.148 0.705 0.225 �0.297Adult 0.431 �0.117 0.535 0.372 0.248 0.309 0.84 0.427 �0.335 0.835 0.294 �0.585



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http://panamath.org


659 Appendix A

660 Average achievement test and IQ scores.

Grade Achievement testand maximumpossible score

Average scoreand SD

Percentage ator above gradelevel proficiency

Average matrixreasoning andvocabularyIQ and SD

First BEAR, 41 30.11, 5.54 89

Second/Third BEAR, 35 28.77, 7.22 73 115.24, 12.28

Fourth/Fifth OCCT Math, 990 854.74, 76.18 100 117, 12.81OCCT Reading, 990 822.35, 61.81 96

Seventh/Eighth OCCT Math, 990 804.47, 55.15 100OCCT Reading, 990 817.94, 62.59 94

710

711 Note. BEAR, Basic Early Assessment of Reading; OCCT, Oklahoma Core Curriculum Tests.

712 References

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755 Price, G. R., Palmer, D., Battista, C., & Ansari, D. (2012). Nonsymbolic numerical magnitude comparison: Reliability and validity756 of different task variants and outcome measures, and their relationship to arithmetic achievement in adults. Acta757 Psychologica, 140, 50–57.758 Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85, 59–108.759 Ratcliff, R. (2002). A diffusion model account of reaction time and accuracy in a two choice brightness discrimination task:760 Fitting real data and failing to fit fake but plausible data. Psychonomic Bulletin & Review, 9, 278–291.761 Ratcliff, R. (2013). Parameter variability and distributional assumptions in the diffusion model. Psychological Review, 120,762 281–292.763 Ratcliff, R. (2014). Measuring psychometric functions with the diffusion model. Journal of Experimental Psychology: Human764 Perception and Performance, 40, 870–888.765 Ratcliff, R., & Childers, R. (2015). Individual differences and fitting methods for the two-choice diffusion model. Decision, 2,766 237–279.767 Ratcliff, R., Love, J., Thompson, C. A., & Opfer, J. (2012). Children are not like older adults: A diffusion model analysis of768 developmental changes in speeded responses. Child Development, 83, 367–381.769 Ratcliff, R., & McKoon, G. (2008). The diffusion decision model: Theory and data for two-choice decision tasks. Neural770 Computation, 20, 873–922.771 Ratcliff, R., Thapar, A., & McKoon, G. (2001). The effects of aging on reaction time in a signal detection task. Psychology and Aging,772 16, 323–341.773 Ratcliff, R., Thapar, A., & McKoon, G. (2003). A diffusion model analysis of the effects of aging on brightness discrimination.774 Perception & Psychophysics, 65, 523–535.775 Ratcliff, R., Thapar, A., & McKoon, G. (2004). A diffusion model analysis of the effects of aging on recognition memory. Journal of776 Memory and Language, 50, 408–424.777 Ratcliff, R., Thapar, A., & McKoon, G. (2006). Aging and individual differences in rapid two-choice decisions. Psychonomic Bulletin778 & Review, 13, 626–635.779 Ratcliff, R., Thapar, A., & McKoon, G. (2010). Individual differences, aging, and IQ in two-choice tasks. Cognitive Psychology, 60,780 127–157.781 Ratcliff, R., Thapar, A., & McKoon, G. (2011). Effects of aging and IQ on item and associative memory. Journal of Experimental782 Psychology: General, 140, 464–487.783 Ratcliff, R., Thompson, C. A., & McKoon, G. (2015). Modeling individual differences in response time and accuracy in numeracy.784 Cognition, 137, 115–136.785 Ratcliff, R., & Tuerlinckx, F. (2002). Estimating the parameters of the diffusion model: Approaches to dealing with contaminant786 reaction times and parameter variability. Psychonomic Bulletin & Review, 9, 438–481.787 Ratcliff, R., Van Zandt, T., & McKoon, G. (1999). Connectionist and diffusion models of reaction time. Psychological Review, 106,788 261–300.789 Sasanguie, D., Defever, E., Van den Bussche, E., & Reynvoet, B. (2011). The reliability of and the relation between non-symbolic790 numerical distance effects in comparison, same–different judgments, and priming. Acta Psychologica, 136, 73–80.791 Sekuler, R., & Mierkiewicz, D. (1977). Children’s judgments of numerical inequality. Child Development, 48, 630–633.792 Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking.New York: Oxford University Press.793 Thapar, A., Ratcliff, R., & McKoon, G. (2003). A diffusion model analysis of the effects of aging on letter discrimination. Psychology794 and Aging, 18, 415–429.795 White, C. N., Ratcliff, R., Vasey, M. W., & McKoon, G. (2010). Using diffusion models to understand clinical disorders. Journal of796 Mathematical Psychology, 54, 39–52.

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