Not all numbers are equal: preferences and biases among children and adults when generating random sequences

Towse, John N.; Loetscher, Tobias; Brugger, Peter

doi:10.3389/fpsyg.2014.00019

ORIGINAL RESEARCH article

Front. Psychol., 23 January 2014

Sec. Developmental Psychology

Volume 5 - 2014 | https://doi.org/10.3389/fpsyg.2014.00019

Not all numbers are equal: preferences and biases among children and adults when generating random sequences

$\r\nJohn N. Towse*$ John N. Towse^1*

Tobias Loetscher²

Peter Brugger^3,4

¹Centre for Research in Human Development and Learning, Department of Psychology, Lancaster University, Lancaster, UK
²School of Psychology, Flinders University, Adelaide, Australia
³Department of Neurology, University Hospital Zurich, Zurich, Switzerland
⁴Zurich Center for Integrative Human Physiology, University of Zurich, Zurich, Switzerland

We investigate the number preferences of children and adults when generating random digit sequences. Previous research has shown convincingly that adults prefer smaller numbers when randomly choosing between responses 1–6. We analyze randomization choices made by both children and adults, considering a range of experimental studies and task configurations. Children – most of whom are between 8 and 11 years – show a preference for relatively large numbers when choosing numbers 1–10. Adults show a preference for small numbers with the same response set. We report a modest association between children’s age and numerical bias. However, children also exhibit a small number bias with a smaller response set available, and they show a preference specifically for the numbers 1–3 across many datasets. We argue that number space demonstrates both continuities (numbers 1–3 have a distinct status) and change (a developmentally emerging bias toward the left side of representational space or lower numbers).

Introduction

There is a strong and venerable interest in mathematical cognition and its development. Out of a large body of empirical work, several important and richly developed theoretical accounts of number processing exist, including for example, the triple-code model (Dehaene and Cohen, 1995) and the encoding complex approach (Campbell, 1994). Such theories make important predictions for mathematical processing in particular. Yet they rest on a sound understanding of how numerical values are represented. For example, numbers involve core mathematical properties, such that with the highly familiar example of a ruler, each value is equally distanced from others. Yet it need not be the case that in psychological space, the same metrics – or internal spaces – apply. In the present paper, we demonstrate that indeed not all numbers are equal; some values are more easily accessible than others, for children and adults. Although number preferences have previously been demonstrated for adults, we show important differences – as well as continuities – in biases exhibited across development. At the same time, we will present data that show preferences are not fixed or immutable. In particular, the “attractiveness” of numbers is not simply an inherent feature of those numerical representations. We will show that preferences need to be understood with reference to the presence or absence of numerical neighbors, as well as the researchers’ decision about how to label numerical sets.

An interesting test case for exploring the properties of number representations comes from random number generation tasks. There is a very extensive literature on the psychological interpretations of attempts to produce random choices. Reviewing the work at the time, Brugger (1997) considered almost 300 research papers, demonstrating clear effects from task difficulty, and performance changes as a function of mental health status. Much work has been conducted since that review, making a comprehensive summary impractical here. Suffice to say that different response choices have been studied, including letters of the alphabet, temporal gaps, and spatially arranged keys. These production formats affect performance (Towse, 1998; Vandierendonck, 2000; Vicario, 2012), yet number sequences form a convenient, popular and relatively well-understood choice set. This random number generation task is typically used as a measure of executive functions, stressing for example the capacity to inhibit prepotent sequences (such as neighboring values) and repetition avoidance (the reluctance to reuse numbers quickly) – see Towse and Neil (1998) for indications of the factor structure underlying performance, and Miyake et al. (2000) for a seminal model of executive functioning that incorporates data from a random generation task.

Typically, random generation research dwells on the sequential structure in responses, and/or the relative accessibility of the response set. Insofar as participants in the random number generation task are making repeated number selections, their response choices might serve additional functions. First, they can potentially reveal numerical preferences independently of randomization quality, and second, they may contribute to an understanding of the potential links between numbers and mental space. With respect to the first point, Loetscher and Brugger (2007) re-analyzed data from 16 experiments in which adults completed a Mental Dice Task (MDT), random generation of 66 verbal responses involving the numbers 1–6. Their analysis of the distribution of response choices showed a systematic bias; participants showed a preference for small numbers (i.e., responses “1,” “2,” or “3”). Moreover, this small number bias (SNB) was quantitatively present in every dataset, and was significant for six of the individual datasets. The average surplus of small numbers was 0.68, quantitatively small but highly significant overall. Di Bono and Zorzi (2013) have recently replicated the SNB with a verbal random number task involving the numbers 1–9, while Vicario (2012) has reported data on a random keypress task requiring 90 response choices among all eight fingers mapped onto eight keys. Vicario reported overall a right hand (i.e., larger value) preference, too.

Loetscher and Brugger (2007) were unable to distinguish between several potential explanations of the SNB. However, one suggestion is that a preference for smaller numbers may reflect a combination of (a) the spatial nature of number representations (left to right in going from small to large) and (b) the biologically rooted left-sided exploration bias in space (Jewell and McCourt, 2000). Several sources of evidence point to the spatial coding of internal number representations. These include the spatial numerical association of response codes (SNARC) effect (see Dehaene et al., 1993; Wood et al., 2008; Fischer and Brugger, 2011; Patro and Haman, 2012), magnitude classification tasks (Moyer and Landauer, 1967), the impact for number processing of neuropsychological damage to areas involved in spatial processing (Zorzi et al., 2002; Loetscher et al., 2010) and research on children’s linear and non-linear number line estimations (Siegler and Opfer, 2003).

With respect to the second issue – the link between number preferences in random generation and mental space – Loetscher and Brugger (2007) also established that individual SNBs were correlated with leftward deviations in number line bisection and in some participants, with orientation biases in real space. Moreover, SNB magnitude can be affected by verbal or spatial attention load, and for example left head turns result in the generation of smaller numbers (Loetscher et al., 2008; see also Takio et al., 2013; for evidence that attentional load affects spatial biases in children). Finally, research focusing on the mental number line often suggests some element of spatial compression such that smaller numbers have greater weighting (Banks and Hill, 1974; Siegler and Opfer, 2003).

Two key issues from Loetscher and Brugger (2007) motivate the current analyses. First, they explicitly identified developmental data as important, in order to investigate whether number preferences are acquired through gradual experience or not. For example Wood et al. (2008) suggested the SNARC effect is not reliably found before approximately 9.5 years of age (see also van Galen and Reitsma, 2008). We are not aware of any published data on number biases in children’s random number generation, and thus we investigate children’s task performance in the present paper.

Second, Loetscher and Brugger acknowledged that their analyses were based on the MDT in which there were always six responses. Consequently, it is not known whether or how number preferences are shaped by particular task configuration. Although we have noted the use of different response choices in other work, the impact of this variable has not been considered in any detail. For example, in the context of studying number preferences among digits 1–6, their label “small” might be used in an absolute sense (i.e., there is something special about the first three digits regardless of the range of numbers being considered) or in a relative sense (i.e., for a given range, numbers smaller than the median as opposed to numbers larger than the median). To address these questions requires data from tasks that involve different response set ranges. We consider such data here.

We investigate number preferences through secondary analysis of several children and adult datasets involving random number generation experiments. These involve a suite of experimental manipulations, including the variation of response set size. Rather than amalgamating all possible datasets straight away, we initially describe a complementary set of studies involving response choices 1–10. We subsequently describe adult performance also using this response range, and then children using responses 1–7. This sectioning permits a focus on a range of important issues.

Analysis 1: Number Preferences Among Children

We took advantage of the opportunity to (re) analyze existing random generation data from children. The most common procedure is to ask children to choose among 10 alternatives (i.e., respond with numbers 1–10) and so we begin with this format.

Materials and Methods

Participants

The present analysis comprises 225 children who verbally produced random sequences using responses between 1 and 10 (at least as part of their task session; n.b. This allowed us to include some extra children in analyses who had to be excluded from the original paper because of missing or problematic data in other conditions). These datasets are described in Table 1, ordered by mean sample age.

TABLE 1

TABLE 1. Small/large number biases in children’s number preferences.

The youngest group was reported as Experiment 3 in Towse and Mclachlan (1999a), with datasets from Towse and Mclachlan (1999b) and Towse and Mclachlan (1999a) [originally described as Experiment 1 (the slow condition) and Experiment 2 respectively]. Finally, we analyze data from an unpublished study of secondary school children (part of an experiment manipulating response speed among children).

The age range of the composite sample varies considerably, between 4 years 11 months and 16 years 1 month (see Table 1 for age ranges). All children produced 70 responses in a sequence except for the final-mentioned group who produced 75 responses. Responses were cued by an auditory tape comprising a systematic sequence of computer beeps. The inter response interval was 2.5 s for all datasets, except for the youngest group, where the interval was 1.5 s. Responses were monitored by the Experimenter who recorded any omissions or errors (i.e., responses outside the range), data that are not considered here.

Results

Response preferences across the corpus are reported in Figure 1. There was overall, a bias toward larger numbers (i.e., the frequency of response choices 1–5 < 6–10). There was a mean excess of 2.03 larger numbers relative to smaller numbers, t(224) = 3.38, p = 0.001, η² = 0.049. Table 1 describes the number preference data for each study cohort separately. Every dataset produced, descriptively, a large number bias (LNB), albeit only in one case was this significant at the individual study level. Notably, that involved the youngest age group (mean age 6;5) whilst the smallest bias occurred among the oldest group (mean age 14;1). Across all these data, we found a small but reliable correlation between children’s age and the size of the SNB, r(223) = 0.149, p = 0.025.

FIGURE 1

FIGURE 1. Distribution of response choices when attempting to generate random sequences. (A) Aggregated data across datasets from children when selecting among the numbers 1–10. (B) Adult preferences when selecting among the numbers 1–10, with written and verbal response modes. (C) Children’s preferences when selecting among the numbers 1–7. Horizontal lines represent chance performance.

Therefore, in contrast to results found with adult samples, children overall show a reliable LNB, not an SNB. However, previous analyses have used the MDT in which there were six response alternatives, and thus small numbers were defined as 1, 2, and 3. We therefore investigated specifically whether numbers 1–3 were used preferentially. We refer to this as a first numbers bias (FNB), potentially reflecting a preference for the earliest acquired numbers. Indeed, the first three numbers were produced more often than the average of all ten response alternatives (mean FNB = 0.373 items), but this was not a statistically reliable effect, t(224) = 1.51, p = 0.132, η² = 0.010. None of the individual study analyses showed significant biases either, though all trends were in the direction of an FNB. A third, alternative analytic approach is to follow the MDT procedure literally and compare the frequency of responses 1–3 (M = 21.8) with 4–6 (M = 20.0; thereby ignoring number choices 7 through 10). This produced a strong FNB, t(224) = 4.32, p < 0.001, η² = 0.075.

In sum, almost paradoxically, we find both a preference for numbers 1–3, and a preference for the larger-than-the-median numbers. Clearly this emphasizes the importance of the operational definition of “small.” It also demonstrates that the SNB reported previously is not a universal preference, once data from children are considered.

Discussion

A priori, we anticipated an SNB, or perhaps the developmental emergence of an SNB from a baseline of no preference. Such an outcome would be consistent with the acquisition of number space correspondences as children become increasingly familiar with numerical material. Instead, what we actually observed was that when choosing between numbers 1–10, children showed an LNB, where large is defined by a median-split with respect to the range of response options. We also found a weak but systematic developmental trend for the emergence of small number preferences. In addition we obtained evidence that number preferences vary with the way magnitude contrasts are constructed. If “small” is defined instead with reference only to the numbers 1–3, then the picture changes yet again, with more evidence for a bias toward these first, or early acquired numbers.

The current analysis has focused on response choices with the numbers 1–10 since this allows for analysis of the largest aggregated set of data. Given the apparently contrasting findings to those reported from adults by Loetscher and Brugger (2007; see also Di Bono and Zorzi, 2013), we next report corresponding random generation data from adults who also used numbers 1–10. This allows us more clearly to distinguish developmental differences from task configuration effects. We chose to analyze two datasets from Towse and Valentine (1997). One involved written responses, a further study elicited verbal sequences.