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

longer a reliable predictor of estimation performance [R = .99, adj. R 2 = .97, F(2, 15) =
295.11, p < .001; see Table A]. On the other hand, when starting the regression analysis with
the two-linear predictor and then adding the logarithmic predictor the latter did not account
for a reliable portion of additional variance [R = .99, adj. R 2 = .97, F(1, 16) = 553.59, p <
.001, b two-lin = .99, p < .001, b log = - .35, p = .18]. Please note that the results for the Germanspeaking
sample were identical.
Table A: Results of the first logarithmic then two-linear forward regression
analysis for Italian-speaking children
Standardized Change in
Predictor
B
t p
b
R²
Logarithmic - .36 - .35 .93 1.42 .18
Two-linear 1.35 1.33 .05 5.43 .001
(b) Identifying the optimal break-point in a two-linear regression may provide
additional information. Only if the optimal break-point falls near the theoretically proposed
break-point the two-linear model seems appropriate. As can be seen from Figure 2 adjusted R 2
for the two-linear model is descriptively larger than that for the logarithmic model.
Furthermore, the optimal break-point of 9.57 for the Italian-speaking as well as 11.33 for the
German-speaking sample was indeed very close to the hypothesized break-point of 10.
(c) Finally, running separate analyses for one- and two-digit numbers may also be of
particular interest for two main reasons. First, when assuming each of these two
representations to be linear with a fixed breakpoint at 10 instead of an overall logarithmic
representation of number magnitude such analyses reflect a crucial test of the predictions of
the two-linear model. On the other hand, when assuming an overall logarithmic representation
not differentiating between single- and two-digit numbers separate analyses for each of these
intervals should nevertheless reveal that the estimates are fitted best by a logarithmic function.
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