The problem with percentages.

A great many students at a major research university make basic conceptual mistakes in responding to simple questions about two successive percentage changes. The mistakes they make follow a pattern already familiar from research on the difficulties that elementary school students have in coming to terms with fractions and decimals. The intuitive core knowledge of arithmetic with the natural numbers makes learning to count and do simple arithmetic relatively easy. Those same principles become obstacles to understanding how to operate with rational numbers.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

What counts? Visual and verbal cues interact to influence what is considered a countable thing.

Many famous paintings illustrate variations in what we here dub "referential depth." For example, paintings often include not only portrayals of uniquely referenced items, but also reflections of those items in mirrors or other polished surfaces. If a painting includes both a dancer and that dancer's reflection in a mirror, are there one or two dancers in the painting? Although there are two images of a dancer, both images reference the exact same dancer. Consequently, counting both may seem to violate the constraint against double counting (Gelman & Gallistel, 1978). This illustrates that determining which things "count" in a given context may not be straightforward. Here we used counting tasks paired with illustrations that manipulated referential depth to investigate the conceptual, perceptual, and language variables that may influence whether a "thing" is a "countable thing." Across four experiments, 316 participants counted items in displays that included both foreground items and items placed inside mirrors, picture frames, and windows. Referential depth and frame boundaries both influenced counting: For one thing, participants were more likely to count items contained by windows than by picture frames or mirrors. Moreover, items in mirrors were rarely counted unless they were interpreted as reflections of items "off screen." Also, the items contained inside windows were sometimes (~10% of trials) excluded from the counts, when counting them would require crossing frame boundaries. We concluded that conceptual and perceptual contexts both influence people's decisions about the physical boundaries of the to-be-counted set and which items within these boundaries are countable.

Young infants have biological expectations about animals.

What are the developmental origins of our concept of animal? There has long been controversy concerning this question. At issue is whether biological reasoning develops from earlier forms of reasoning, such as physical and psychological reasoning, or whether from a young age children endow animals with biological properties. Here we demonstrate that 8-mo-old infants already expect novel objects they identify as animals to have insides. Infants detected a violation when an object that was self-propelled and agentive (but not an object that lacked one or both of these properties) was revealed to be hollow. Infants also detected a violation when an object that was self-propelled and furry (but not an object that lacked one or both of these properties) either was shown to be hollow or rattled (when shaken) as although mostly hollow. Young infants' expectations about animals' insides may serve as a foundation for the development of more advanced biological knowledge.

Visual nesting impacts approximate number system estimation.

The approximate number system (ANS) allows people to quickly but inaccurately enumerate large sets without counting. One popular account of the ANS is known as the accumulator model. This model posits that the ANS acts analogously to a graduated cylinder to which one "cup" is added for each item in the set, with set numerosity read from the "height" of the cylinder. Under this model, one would predict that if all the to-be-enumerated items were not collected into the accumulator, either the sets would be underestimated, or the misses would need to be corrected by a subsequent process, leading to longer reaction times. In this experiment, we tested whether such miss effects occur. Fifty participants judged numerosities of briefly presented sets of circles. In some conditions, circles were arranged such that some were inside others. This circle nesting was expected to increase the miss rate, since previous research had indicated that items in nested configurations cannot be preattentively individuated in parallel. Logically, items in a set that cannot be simultaneously individuated cannot be simultaneously added to an accumulator. Participants' response times were longer and their estimations were lower for sets whose configurations yielded greater levels of nesting. The level of nesting in a display influenced estimation independently of the total number of items present. This indicates that miss effects, predicted by the accumulator model, are indeed seen in ANS estimation. We speculate that ANS biases might, in turn, influence cognition and behavior, perhaps by influencing which kinds of sets are spontaneously counted.

An encounter frequency account of how experience affects likelihood estimation.

When making judgments, people often favor information received from a few individual sources over large-sample statistical data. Individual information is usually acquired piece by piece, whereas statistical information combines many observations into a single summary. We examined whether this difference in the frequency of encounters affects how data are weighted. In two experiments, subjects read statistical information indicating an event to be rare and contrasting information from individual cases suggesting the event to be common. We controlled whether the individual cases were summarized into a single summary like statistical information, or presented serially, case by case. Subjects' estimates of event frequencies were higher when the individual cases were presented in serial, rather than summarized, format. A third study demonstrated that subjects treat each data sample as an instance, and do not weight according to sample size. These results support the conclusion that people weight information according to encounter frequency.

The generative basis of natural number concepts.

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the arithmetic principle, supports exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes.

Nonverbal arithmetic in humans: light from noise.

Animal and human data suggest the existence of a cross-species system of analog number representation (e.g., Cordes, Gelman, Gallistel, & Whalen, 2001; Meeck & Church, 1983), which may mediate the computation of statistical regularities in the environment (Gallistel, Gelman, & Cordes, 2006). However, evidence of arithmetic manipulation of these nonverbal magnitude representations is sparse and lacking in depth. This study uses the analysis of variability as a tool for understanding properties of these combinatorial processes. Human subjects participated in tasks requiring responses dependent upon the addition, subtraction, or reproduction of nonverbal counts. Variance analyses revealed that the magnitude of both inputs and answer contributed to the variability in the arithmetic responses, with operand variability dominating. Other contributing factors to the observed variability and implications for logarithmic versus scalar models of magnitude representation are discussed in light of these results.

Intuitive t tests: lay use of statistical information.

Normatively, a statistical pairwise comparison is a function of the mean, standard deviation (SD), and sample size of the data. In our experiment, 203 undergraduates compared product pairs and judged their confidence thatone product was better than the other. We experimentally manipulated (within subjects) theaverage productratings, the number of raters (sample size), and the SD of the ratings. Each factor had two levels selected, so that the same change in statistical power resulted from moving from the low to the high level. We also manipulated (between subjects) whether subjects were given only the product rating data as summarized in a statistical format orthe summariesplus the raw ratings. Subjects gave the most weight to mean product ratings, less weight to sample size, and very little weight to SD. Providing subjects with raw data did not increase their use of sample size and SD, as predicted.

Sometimes area counts more than number.

Using an interference paradigm, we show across three experiments that adults' order judgments of numbers, sizes, or combined area of dots in pairs of arrays occur spontaneously and automatically, but at different speeds and levels of accuracy. Experiment 1 used circles whose sizes varied between but not within arrays. Variation in circle size interfered with judgments of which array had more circles. Experiment 2 used displays in which circle size varied within and between arrays. Between-array differences in the amount of "circle stuff" (area occupied by circles) interfered with judgments of number. Experiment 3 examined whether variation in number also interferes with judgments of area. Interference between discrete and continuous stimulus dimensions occurred in both directions, although it was stronger from the continuous to the discrete than vice versa. These results bear on interpretations of studies with infants and preschoolers wherein subjects respond on the basis of continuous quantity rather than discrete quantity. In light of our results with adults, these findings do not license the conclusion that young children cannot represent discrete quantity. Absent data on attentional hierarchies and speed of processing, it is premature to conclude that infant and child quantity processes are fundamentally different from that of adults.

Number and language: how are they related?

Does the ability to develop numerical concepts depend on our ability to use language? We consider the role of the vocabulary of counting words in developing numerical concepts. We challenge the 'bootstrapping' theory which claims that children move from using something like an object-file - an attentional process for responding to small numerosities - to a truly arithmetic one as a result of their learning the counting words. We also question the interpretation of recent findings from Amazonian cultures that have very restricted number vocabularies. Our review of data and theory, along with neuroscientific evidence, imply that numerical concepts have an ontogenetic origin and a neural basis that are independent of language.

Language and the origin of numerical concepts.

Reports of research with the Pirahã and Mundurukú Amazonian Indians of Brazil lend themselves to discussions of the role of language in the origin of numerical concepts. The research findings indicate that, whether or not humans have an extensive counting list, they share with nonverbal animals a language-independent representation of number, with limited, scale-invariant precision. What causal role, then, does knowledge of the language of counting serve? We consider the strong Whorfian proposal, that of linguistic determinism; the weak Whorfian hypothesis, that language influences how we think; and that the "language of thought" maps to spoken language or symbol systems.