Consistency of Response Patterns in Different Estimation Tasks

Abstract

The current study aimed at addressing two issues concerning children's estimation performance: (1) to investigate whether the log-to-linear framework or the proportional judgment framework provided a better explanation of children's estimation patterns, and (2) to examine the consistency of response patterns in different estimation tasks. A sample of 179 Chinese first graders was assessed on their arithmetic performance and estimation skills (including numerosity naming, numerosity production, and number line). The log-to-linear framework was suggested to provide a better framework in explaining children's estimation patterns. Under this framework, we identified both common features and uniqueness of children's response patterns in different estimation tasks. Furthermore, different estimation skills uniquely contributed to children's arithmetic performance. The theoretical and practical implications of the findings are discussed.

Notes

¹ In cases in which the responses severely deviated from the target, the experimenter would provide some hints, such as “25 is a relatively small number, so it should be on the left side,” in the number line task. Exact answers were not given to the participants. We further tested whether providing this kind of feedback would affect participants’ performance. If the feedback has an effect, we would expect the items closer to the landmarks (distance from landmarks ≤ 6, where landmarks refer to 25, 50, and 75) to be more accurate than the items farther away from the landmarks (distance from landmarks > 6). This was not the case. The close-landmark items were as accurate as the far-landmark items, t(165) = –0698, p = .486), suggesting that this general feedback did not affect participants’ performance in any significant way.

² The number of outliers excluded for each variable: R²_lin or R²_exp in numerosity naming = 3; R²_lin or R²_ln in numerosity production = 2; R²_lin or R²_ln in number line = 5; Corsi block = 1; backward digit span = 1. Another participant was excluded because her IQ score was less than 80.

³ In the LOOCV procedure, a regression model was constructed using all but one data point. This model was then used to fit the remaining data point to see if the model was accurate in predicting new data. Here, the mean absolute percent error (MAPE; lower value indicates better fit) of each participant for both functions in the three estimation tasks was calculated. The results suggested a lower average logarithmic MAPE for the numerosity production task (compared with the average linear MAPE) and lower average linear MAPEs for the numerosity naming and number line tasks (compared with the average exponential/logarithmic MAPEs).

⁴ For example, at a 5% cutoff, a participant would be classified as having a logarithmic response pattern only when his/her response pattern in that task fit the logarithmic function by at least 5% more than the linear function.

⁵ For all tasks, when participants were classified as a logarithmic/exponential fit, the average logarithmic/exponential MAPE was lower than the linear MAPE (except for numerosity naming, in which the average exponential MAPE was lower than the average linear MAPE only when a more stringent cutoff was used [i.e., 5%]). On the other hand, if the participants were classified as a linear fit, the average linear MAPE was lower than the logarithmic MAPE. The cross-validation results therefore provided further support for our classification.