The influence of the place-value structure of the Arabic number ...

two-linear model with one linear representation of one-digit numbers and another linear
representation of two-digit numbers. In the data of Moeller and colleagues (2009a) both
models captured the data well (i.e., the logarithmic as well as the two-linear with a fixed
breakpoint at 10). However, in direct comparison and some critical tests the two-linear model
outperformed the logarithmic model reliably in terms of descriptive adequacy (i.e., R 2 ) in a
German-speaking children sample. From this finding, Moeller et al. (2009a) conclude that the
spatial coding of numerical magnitude along the MNL may not develop from logarithmic to
linear with increasing age and experience as previously suggested by Siegler and colleagues
(e.g., Opfer & Siegler, 2007; Siegler & Booth, 2004). Instead, the developmental change may
involve the mastery of the place-value structure of the Arabic number system. As observed by
Moeller and co-workers (2009) as well as by many others (e.g., Siegler & Opfer, 2003; Opfer
& Siegler, 2007) children seem to systematically overestimate the space reflecting the
position of one-digit numbers towards the right in the beginning (i.e., locating 7 at the actual
position of approximately 30). However, on average they also adhere to the ordinal order of
the to-be-estimated numbers. Moeller et al. (2009a) interpreted this as indicating that children
already seem to know that, for instance, 40 is somehow larger than 4; nevertheless, they are
not yet capable of correctly representing the ratio of this “larger” complying with the base-10
place-value structure of the Arabic number system (i.e., the distance between 0 and 40 is 10
times as large as the distance between 0 and 4). When the “somehow” in above sentence
means not 10 times larger, but 2 or 3 times larger, this could potentially produce data patterns
which are well fitted by a logarithmic fitting although the underlying data may not be
logarithmic 4 . Thus, it may not be a logarithmic to linear change of children’s magnitude
4 Please note that this argument holds for both directions. Data produced by a logarithmic model can also be well
fitted by a two-linear model, at least when no fixed break points around 10 is assumed as in the model of
Ebersbach et al. (2008). Thus, descriptive adequacy alone cannot prove that the underlying data are also
produced by either model fitting the data (cf. Myung & Pitt, 1997, see also Appendix A for further discussion
of this point).
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