Fig 1.
Schematic illustration of the numerical development views.
In the unique-representation view, the ANS is the basis of mathematics and is activated by both symbolic and non-symbolic numbers. In the dual-representation view, non-symbolic numbers activate the ANS and symbolic numbers activate an exact representation of numbers; the latter is the root of mathematical learning and can refine the ANS.
Fig 2.
Exact addition task.
Fig 3.
Illustration of the symbolic (a) and the non-symbolic (b) versions.
Fig 4.
Illustration of the non-symbolic (a and c) and symbolic (b and d) versions. In the first level, children were asked to position a numerosity on a number line from 1 to 10 (a) and, in the second level, on a number line from 1 to 20 (b). Feedback was given after each trial; a mouse looking at the left (c) / an elephant looking at the right indicated whether their estimate was too big/small.
Table 1.
Game level at the end of training for each training game in each group.
Table 2.
Groups’ mean and standard deviations for each variable at T1 and at T2 (%).
Table 3.
Cronbach’s alpha for each of the testing measures at T1 and T2.
Fig 5.
Performance in the collection comparison task.
Illustration of the interaction between ratio and time in the non-symbolic, the symbolic and the control groups. Error bars represent standard errors.
Table 4.
Means ± standard deviations in the non-symbolic number line task.
Fig 6.
Performance in non-symbolic number line.
Illustration of the decrease of PAE from T1 to T2 in each group for small and large numerosities. Error bars depict standard errors.
Fig 7.
Performance in Arabic number and verbal number comparison tasks.
Error bars represent standard errors.
Fig 8.
Performance in non-symbolic and symbolic number line.
Error bars depict standard errors.
Fig 9.
Performance in exact addition.
Interaction between time and group. Error bars represent standard errors.
Table 5.
Training effect sizes (r) in different training studies, including the present study.