1. Knower-Levels and the Acuity of the Approximate Number System James Negen and Barbara W. Sarnecka University of California, Irvine {jnegen,sarnecka}@uci.edu Abstract The approximate number system (ANS) – also known as the analogue magnitude system, the ‘number sense’, and the ‘number module’ – is a cognitive system that provides nonverbal estimates of the number of items in a stimulus. Adults can map these estimates to number words (e.g., Dehaene, 1997). Some theories say that the ANS provides numeric meaning for the number words from the beginning (e.g., Gelman & Gallistel, 1992); others say that cardinal number-word meanings are not initially connected to the ANS (Carey, 2009; Le Corre & Carey, 2007). Knower-levels are a way of classifying children based on what number words they know. 1-knowers knows ‘one’, 2-knowers know ‘one’ and ‘two’, etc., and CP-knowers know how to assign meanings to number words through counting. Wagner and Johnson (2011) argue that the child’s progress at learning the first few number-word meanings is linked to her ANS acuity. Our findings here do not support this conclusion. Methods Results “Can you give Mr. Bunny TWO lemons?” Give-N Task Diagnoses a child’s Knower-Level (i.e., what number words the child knows) Experimenter asks the child to give a certain number of items to a stuffed animal Requested number starts small, increases with correct answers and decreases with incorrect ones Example to left: a child, asked to give two lemons, puts three on a plate (left frame) and slides them to the animal (right frame) “Which side has MORE dots?” Set-Comparison Task Measures a child’s ANS Acuity Experimenter asks the child “Which side has MORE dots?” First ratios are very easy (3:1). The child may not go on until 8 very easy trials in a row are correct. Ratios get progressively harder (until 10:9) Half are area-correlated, half are area-anticorrelated (example left) so that non-numeric cues are not useful Feedback given after every trial (e.g., “Yes, that’s right.”) Background Discussion The set-comparison task measures ANS acuity. Age is a significant predictor of set-comparison performance (p <.001), as is knower-level (p =.018). However, knower-level is not correlated with set- comparison performance when partialled for age, r(55) =.091, p =.251. We have not found a statistically significant relationship between ANS acuity and knower- level when age is controlled. There are at least three possible interpretations of these results: First, these findings could point away from the conclusions of Wagner & Johnson (2011), who suggest that ANS acuity predicts number-word knowledge. Our method of measuring ANS acuity may explain the difference in results: We used 20-100 dots in each set (as apposed to ¼ of trials having only 1-4 items), excluding the use of parallel individuation. We had area-correlated and -anticorrelated trials (not just area-equated trials), so overall above-chance performance must rely on numeric perceptions. Studies that just use area-equated trials can’t distinguish a child with high ANS acuity from a child that just usually picks the smaller dots. Our participants had to demonstrate understanding of the instructions before training ended, and continued to receive feedback. This insures that any observed effects are not driven by differences in understanding the instructions. Second, since most participants have been 3-knowers or higher, it is possible that more 1-knowers and 2-knowers are needed to see the effect. Finally, since 46/58 children performed better on area-correlated trials than area- anticorrelated trials, it is possible that a proper analysis needs to separate the child’s ANS acuity from her ability to ignore irrelevant information. Special thanks to Jessica Sullivan and David Barner for providing an alternative version of the Set-Comparison Task for us to adapt. 46 children performed better on area-correlated trials while 10 performed better on area-anticorrelated trials (p <.001, binomial). The average absolute difference was 27%. This may indicate a need for an analysis method that estimates acuity separately from bias towards the larger (or smaller) set. Note that this occurred despite the fact that all children got 8 cards correct in a row at the 3:1 ratio, which included 4 area- anticorrleated trials. Also, children under 5 years old performed above chance on area-anticorrelated trials (1616/2880, p <.001), contrasting with Soltész, Szűcs & Szűcs (2011). Here we present a study of 58 children (2;4 to 7;6) that examines the relationship between number-word knowledge and the acuity of the approximate number system (ANS). In contrast to other recent findings (Wagner & Johnson, 2011), we find no significant relationship between ANS acuity and number- word knowledge when age is controlled. This suggests that the number-word learning process is not driven by improvements in ANS acuity. Future work on this study will (1) continue to collect more data and (2) develop methods of analysis that accounts for the child’s impulse to simply pick the set with greater area.