Psychophysics of Numerical Representation

Awidely agreed method to assess children’s spatial representation of number magnitude along an assumed mental number line is the so-called number line estimation task. In this task, children are required to indicate the position of a given number (e.g., 37) upon a (number) line with marked end points (e.g., ranging from 0 to 100). Performance of older children and adults in this task was repeatedly observed to be linear (e.g., Siegler & Opfer, 2003), implying that participants considered the distance between 0 and 40 to be 10 times the distance between 0 and 4 – thereby acknowledging the base-10 structure of the Arabic number system and equidistance between adjacent numbers. However, considering the 0–100 range, children initially overrepresented the spread of the relatively smaller numbers within the range by placing them too far to the right, for instance positioning the number 9 at about the position of 40 (cf. Moeller, Pixner, Kaufmann, & Nuerk, 2009). Typically, this data pattern was accounted for by a logarithmic model, taken to indicate an initially logarithmic representation of number magnitude in children (e.g., Opfer & Siegler, 2007; Siegler & Booth, 2004; Siegler & Opfer, 2003). With increasing age and experience children’s estimates get more and more precise, finally resulting in a quite accurate one-to-one correspondence between the number to-be-positioned and its actual position on the number line, described best by a linear function and suggested to index the end point of a logarithmic to linear development of children’s (spatial) number magnitude representation (cf. Opfer & Siegler, 2007). However, this view was questioned recently. On the one hand, Ebersbach, Luwel, Frick, Onghena, and Verschaffel, (2008) suggested a two-linear instead of a logarithmic representation of number magnitude from 0 to 100, with the breakpoint between the two linear segments reflecting the point up to which children are sufficiently familiar with the numbers. On the other hand, Moeller and colleagues (Helmreich et al., in press; Moeller, Pixner et al., 2009) picked up on the idea of a two-linear model. Yet, different from Ebersbach et al. (2008) they proposed a fixed breakpoint at 10 indicating representational differences between singleand two-digit numbers. The authors interpret this to reflect an impact of the place-value structure of the Arabic number system on the development of number magnitude representation. The proposal of a two-linear model was motivated by earlier findings from a variety of tasks suggesting that two-digit number magnitude may not be processed holistically (i.e., as an integrated entity) but decomposed into tens and units (see Moeller, Nuerk, & Willmes, 2009; Nuerk, Weger, & Willmes, 2001; see Nuerk & Willmes, 2005; Nuerk, Moeller, Klein, Willmes, & Fischer, 2011 (this issue) for reviews). The debate about logarithmic versus two-linear representations of number magnitude has both theoretical and empirical aspects. In the first and major part of this opinion section, we describe which methods can be employed to distinguish between the models empirically and apply these methods to the data of two recent studies by Helmreich