Straightening Up: Number line Estimates Shift from Log to Linear with Additional Information

Straightening Up: Number Line Estimates Shift from Log to Linear with Additional Information Percival G. Matthews (pmatthew@nd.edu) Dana Chesney (Dana.Chesney.3@nd.edu) University of Notre Dame, Department of Psychology, 118 Haggar Hall, Notre Dame, IN 46556 Abstract It has been suggested that a developmental log-to-linear shift in children’s performance on number line estimation tasks is diagnostic of their underlying representations of numerical magnitude (Siegler & Opfer, 2003). However, in the study presented herein, we were able to induce a similar log-to- linear shift on number line estimation tasks among adults by manipulating their familiarity with the numbers we used as stimuli. We offer this evidence as an existence proof that differences in performance on number line estimation tasks may not necessarily be indicative of fundamental differences in the formats of people’s underlying numerical magnitude representations. Rather, they may be diagnostic of differences in people’s understandings of what magnitudes are represented by symbolic numbers. Keywords: Number line, Number, Numeracy, Estimation Approximate number system Numerical estimation is an important part of mathematical cognition for both children and adults. Indeed, numerical estimation seems to play a central role in a wide range of mathematical activities (see Siegler & Booth, 2005 for a review). In recognition of its importance, Siegler and colleagues have focused on the ability to estimate numerical magnitudes as a key indicator of number sense in developmental studies (Siegler & Booth, 2004; Siegler & Opfer, 2003) Perhaps the most well known of these numerical estimation tasks is the number line estimation task. This task requires that participants estimate the location of a number on a line with numerical anchors at each end. This task not only involves magnitude estimation, but also the ability to translate between symbolic numbers and mental magnitudes. Siegler and colleagues have helped illuminate a typical developmental trend observed on such number line estimation tasks: young children show a tendency to compress numbers logarithmically, whereas adults do not (e.g., Siegler & Booth, 2004; Siegler & Opfer, 2003). That is, when asked to mark the position where a number n is on the line, children typically place the mark at a spot approximated by log(n) instead of n. With development, these estimates become more linear (Siegler & Opfer, 2003). Importantly, the linearity of children’s magnitude estimates on these tasks is correlated with a wide range of numeracy measures, including counting ability, number naming, digit magnitude comparison, and achievement test scores (e.g., Booth, 2005; Ramani & Siegler, 2008; Siegler & Booth, 2004). According to the log-to-linear shift hypothesis, the logarithmic compression found in children is diagnostic of their underlying representations of numerical magnitude. This view suggests that children’s performance is logarithmic because their underlying mental representations of numerical magnitudes are logarithmically compressed (Dehaene, 1997; Siegler, 2009). Siegler and colleagues further theorize that experience and schooling lead to the development of a linear representation of numerical magnitude. More specifically, they propose that though the logarithmic and linear representations continue to coexist, individuals can learn to invoke the linear representation when it is appropriate (Siegler & Opfer, 2003). Three pieces of evidence appear to support this hypothesis. First, children’s estimates become more linear with schooling and experience. Second, the linearity of children’s estimates predicts a wide range of numeracy measures. These include counting ability, number naming, digit magnitude comparison, and achievement test scores, perhaps indicating that the hypothesized linear ruler is widely applied once it is developed (Siegler & Booth, 2004). Finally,, children’s estimates are more linear within number ranges that are more familiar to them but remain logarithmic on larger, more unfamiliar scales (Siegler & Opfer, 2003). This in particular has been taken as evidence that the logarithmic and linear rulers coexist. The log-to-linear shift hypothesis, however, is not without controversy. A recent critique put forth by Barth and Paladino (2010) raised questions about the interpretation of children’s apparently logarithmic performance. The critique argued that the shift from logarithmic to linear performance on number line tasks is not diagnostic of some basic change in children’s underlying representation of number, but is instead due to knowledge constraints interfering with the default method for completing the task. Barth and Paladino argue that number line estimation cannot properly be seen as a pure numerical estimation task. Rather, such placement tasks are actually a form of proportion judgment task – a task in which ratio between items must be evaluated. Indeed, previous literature in psychophysics has shown that estimation tasks that combine two measures in a complementary fashion such that they sum to a fixed total should be characterized as proportion judgment tasks (e.g., Hollands & Dyre, 2000; Spence, 1990; Stevens, 1957). Thus, because estimating a number’s place on a number line involves both the estimate of that numbers’ placement relative to the zero anchor point and of its complement’s placement relative to the rightmost anchor, the task is