“And so he went back over the sunny hills and down through the cool valleys, to show all his pretty kittens to the very old woman. It was very funny to see those hundreds and thousands and millions and billions and trillions of cats following him.”

1. Introduction

In a 2010 interview, Vice President of the United States, Joe Biden, reported that an unnamed conservative had spent an unprecedented $200 billion in political ads before the midterm election (Hunt, 2010). Were it true, the money would surely have been poorly spent: This amount would have funded the entire operation of the National Science Foundation for 30 years, or alternatively the entire budget of the U.S. military for 4 months. It is not true of course; the figure Mr. Biden intended was $200 million (Shear, 2010). The mistake is a familiar one: Although we all know words like “million,” “billion,” and “trillion,” most of us feel only the vaguest sense of their referents. In this study, we will explore how people come to attribute meaning and value to the magnitudes of numbers with which we largely lack direct experience.

Large numbers (here, roughly those between 105 and 1013) are interesting for both practical and theoretical reasons. First, experimental tasks in economics and psychology frequently ask people to make decisions involving large numbers (see Bohm & Lind, 1993; Cohen, Ferrell, & Johnson, 2002). Moreover, many arenas of public discourse involve large numbers, including debates about evolutionary biology (Dunning, 1997; Meffe, 1994), nanotechnology (Batt, Waldron, & Broadwater, 2008), and the reliability of DNA testing (Koehler, 1997). And, of course, many countries are currently involved in ongoing and heated political conversations about governmental budgets and national economies. The United States is not atypical: there, the budget, the deficit, and the debt are in the low trillions, while most proposed budgets involve changes on the order of millions and billions. Better understanding of how people make sense of numbers in this range has the potential to improve communication in these important arenas.

Large numbers are an excellent example of an abstract system: Magnitudes such as 1 billion are beyond our immediate experience and yet are clearly understood in part through generalizing the concrete process of counting (Carey, 2009; Leslie, Gelman, & Gallistel, 2008). Nearly every adult knows how to count to 1 billion and how to manipulate numbers in this range using basic arithmetic (Skwarchuk & Anglin, 2002), but for most of us, we do not directly work with large numeric magnitudes. Instead, relevant experiences are typically limited to encounters with large number representations through associations with relevant situations (usually by encountering assertions; e.g., that the U.S. budget deficit is $1.4 trillion; Facebook has 700 million users; or the human body has 100 trillion cells).

One way we understand abstractions is by studying the properties of their concrete representations (Barsalou, 1999; Clark, 2006; Kirsh, 2010; Landy & Goldstone, 2007). For instance, Carey (2009) proposes that when learning to count, the memorized count list orients attention to appropriate features of the environment, so that the verbal label “eighteen” cues a learner that there is something that “eighteen” situations have in common. In addition to the simple presence or absence of labels, count lists have structural properties: They are typically stated in sequence, with accompanied rhythmic hand motions, and are constructed on a semiregular linguistic pattern. Here, we investigate how these structural components of symbolic systems impact inferences made by reasoners.

1.1. Structure in the numerals A student learning the English counting system must master several different lists. In addition to the numbers from 1 to 9, for instance, one must learn the teen words and the tens words (10, 20, 30, and so on). The most important count list for our purposes is the short scale, commonly used in the United States and Britain (Conway & Guy, 1996). In this system, 1,000 million is “1 billion.”1 This list “thousand, million, billion, trillion, quadrillion, …” constitutes an effective count list, which after the initial “thousand,” bears an apparent sequential structure and clearly derives from Latin number words. To represent a number in the short scale, one divides the number into an order (e.g., “thousands”) and a numeral phrase that runs from 0 to 999; one can think of this system as a standard positional notation with base 1,000; by analogy, we will treat a representation such as 340,000,000 as having the digit 340 and the place “000,000” or “million.” In British and American numerical grammar, some words combine additively, and others combine multiplicatively (Hurford, 1987, 2007). Forty‐three is an example of the former, as its value just 40 + 3. Seven hundred illustrates multiplicative combination, 7 × 100. Normatively, words in the short scale combine multiplicatively with the immediately preceding term. The short scale is unlike the other standard scales (units and tens words) in two ways. First, like “hundred,” the short scale combines multiplicatively. Second, the magnitudes captured by the short scale increase exponentially rather than linearly. North American students typically learn the short scale up to “trillion” by around seventh grade (Skwarchuk & Anglin, 2002). Substantial research has investigated the struggles of children learning the verbal number system, although most investigations have focused on numbers under 100, with very few explorations of numbers over 1,000, let alone over 1 million (e.g., Baroody & Price, 1983; Gelman & Gallistel, 1978; Siegler & Robinson, 1982; but see Skwarchuk & Anglin, 2002).

1.2. Number line estimation and its relationship to magnitude processing In the number‐line estimation task (Siegler & Opfer, 2003), participants estimate the appropriate location for a presented number on a line with labeled end points. Roughly, children present one of two patterns of behavior on number‐line tasks: Given a particular age range and a particular number range, students either err by compressing the terms toward the high end or correctly place terms close to linearly (Booth & Siegler, 2006). The transition from compressive to linear placement is typically extended, happening at different times for different number ranges. The shift to linear behavior for 0–100 typically occurs between kindergarten and second grade (Siegler & Booth, 2004), for 0–1,000 between second and sixth grade (Siegler & Opfer, 2003), and for ranges up to 100,000 (the largest previously tested) between fourth and sixth grade (Siegler, Thompson, & Opfer, 2009; Thompson & Opfer, 2010). Individual transitions are often quite sharp, with many participants exhibiting single‐trial learning after feedback (Opfer, Siegler, & Young, 2011). Children exhibiting linear number‐line behaviors also typically have better memory for numbers (Thompson & Siegler, 2010), suggesting that number‐line behaviors relate meaningfully with behavior on other numeric tasks. Several theories have been proposed to account for children's number‐line behavior. Opfer et al. (Opfer et al., 2011; Siegler & Opfer, 2003) explain the behavioral transition in terms of a discontinuous shift from logarithmic to linear representations of numeric magnitude. An alternative pair of models, reminiscent of the account pursued here for large number ranges, holds that children develop separate expectations for numbers under 10, and those between 10 and 100 (Moeller, Pixner, Kaufmann, & Nuerk, 2009; Nuerk, Weger, & Willmes, 2001), or, more generally, separate expectations for numbers they know, versus those they do not (Ebersbach, Luwel, Frick, Onghena, & Verschaffel, 2008). Barth and Paladino (2011) analyze number‐line estimation as a proportion judgment task, using a power function to model the compression of large numbers (extending the cyclic power model of Hollands & Dyre, 2000). They then explain the observed behavioral shifts through a postulated continuous shift in proportion estimation and through refinements in children's understanding of larger reference numbers. Although these models are conceptually distinct, both the segmented models of Nuerk et al. and Ebersbach et al. and the variant of the cyclic power model used by Barth and Paladino produce predictions extremely similar to the logarithmic model, making them difficult to distinguish empirically (Young & Opfer, 2011; Opfer et al., 2011). Although our experiments use a different population over a much larger number span, they may illuminate children's behavior by presenting clear‐cut illustrations of adult behaviors relatable to the models that have been applied to children. These relationships are taken up again in the General Discussion.