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

N
60
Distribution of individual breakpoints
III
40
20
0
37
Breakpoint (rounded)
(d) Finally, a correlation analysis of the individual slopes of the two linear or two
logarithmic segments for one- and two-digit numbers may be informative. Under the
assumption of the two-linear model suggested by Moeller et al. (2009a) the slopes of the two
linear segments of the two-linear model should be negatively correlated: the steeper the slope
of the segment for single digit numbers the further to the right are one-digit numbers placed
on the hypothetical number line. Thereby, all two-digit numbers have to be located on a very
limited part of the number line leading to a relatively flat slope for these two-digit numbers.
On the other hand, the slopes of two logarithmic segments meeting at a fixed breakpoint
should be positively correlated. Thereby, evaluating the correlation of the slopes of the two
linear segments for one- and two-digit numbers is a critical test for the applicability of the
two-linear model and the two separate representations of one-and two-digit numbers it
reflects. For both the Italian- as well as the German-speaking sample the correlation between
the slopes of the two linear segments was significantly negative.
It is obvious that both approaches follow the same logic either across items (see
above) or across participants. However, in the latter approach model fit is assessed on an
individual level, thereby, taking into account possible individual differences in development
which are treated as measurement error in the item-based approach. Therefore, we think that
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