Quantifiers are a test case for an interface between psychological questions, which attempt to specify the numerical content that supports the semantics of quantifiers, and linguistic questions, which uncover the range of possible quantifier meanings allowable within the constraints of the syntax. Here we explore the development of comprehension of most in English, of particular interest as it calls on precise numerical content that, in adults, requires an understanding of large exact numerosities (e.g. 23 blue dots and 17 yellow is an instance of “most of the dots are blue”). In a sample of 100 children 2 to 5 years of age we find that: 1) successful most comprehension in cases with two salient subsets is achieved at 3 years-7 months of age, and 2) most comprehension is independent of knowledge of large exact number words; i.e. knowledge of large exact number words is neither necessary, as evidenced by children who understand “most” but not “four”, nor sufficient, as evidenced by children who understand “nine” but not “most”.

Counting ability was measured for each child (e.g. a one-knower knows the meaning of "one" but no other numbers in their countlist). Second, "most"-comprehension was measured for cases of two salient subsets over multiple trials that varied ratio. Children called out a color on each trial. Click play to view a trial.

While “most”-comprehension increased as a function of age, there was no tendency for “most”-comprehension to increase as a function of counting ability once age was controlled for. We found some older non-counters who understood most, and we found some younger full-counters who did not. In this quicktime, each dot is the average percent correct for a single child (n=100). Click play to see it rotate. Notice that the cloud of data for each level of counting ability overlaps. The data cloud for counting level 4 (full-counters) encompasses the full range seen in the clouds for levels 1-3 (non-counters). That is, full-counters are no different from non-counters once age is controlled for.

These same data can be viewed with the regression line drawn through the points. Here the slope of the regression has been determined through 3-D surface modelling (p < .001, r2 = .449). Notice as the cube rotates that the regression plane bisects the data for all 4 levels of counting ability with some points below the regression plane and some above. Also note that the regression plane has no tilt in the dimension of counting ability. These observations reflect that there is no effect of counting ability once age is controlled for.

Lastly, the regression surface can also be modeled as a sigmoid. While our data appear to capture the linear portion of the transition from failure to success at our task, the full range of development must approximate a sigmoid as chance (50%) and perfect performance (100%) will function as boundary conditions. Here we show the mean and upper and lower confidence interval for the slope of the sigmoid while the upper and lower asymptotes have been constrained to be 50% and 100%. If a wider range of ages were run in our task, this sigmoid would be the most likely regression surface.

Halberda, J., Ly, R., Wilmer, J., Naiman, D., & Germine, L. (2012). Number Sense across the lifespan as revealed by a massive internet-based sample. PNAS [Demo]

Spiegel, C., & Halberda, J. Rapid fast-mapping abilities in 2-year-olds. Journal of Experimental Child Psychology (2010). [Demo]

Halberda, J., Mazzocco, M. & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455, 665-668. [Demo]

Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the Number Sense: The Approximate Number System in 3-, 4-, 5-, and 6-Year-Olds and Adults. Developmental Psychology, 44 (5). [Demo]

Halberda, J., Taing, L. & Lidz, J. (2008). The development of “most” comprehension and its potential dependence on counting-ability in preschoolers. Language Learning and Development. [Demo]

Halberda, J., Sires, S.F., & Feigenson, L. (2006). Multiple spatially-overlapping sets can be enumerated in parallel. Psychological Science. 17 (7), 572-576. [Demo]

Halberda, J. (2006). Is this a dax which I see before me? Use of the logical argument disjunctive syllogism supports word-learning in children and adults. Cognitive Psychology, 53(4), 310-344. [Demo]

Nichols, S. & Halberda, J. (in preparation). Modal reasoning in preschoolers. [Demo]

Prospective research assistants should email Tyler Knowlton (tknowlt2@jhu.edu); prospective graduate students should email Dr. Halberda directly (halberda@jhu.edu).

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