The results of number line estimation and base 10 questions were coded to for the next analyses.
Estimation accuracy The 0−100 number line estimations were measured by the 0−20 CM ruler with scales of every 0.2 CM (equals to number 1) and 0.02 CM (equals to number 0.1) on it. The experimenter used the ruler to measure the results of estimated number magnitudes (e.g., 15). The experimenter input both the estimated and actual number magnitudes of each number and calculates percentage of absolute error (PAE) of each estimation result. For instance, if a student was asked to estimate 25 on a 0−100 number line but mark at the location of 15. PAE= |((15-25))/100| × 100%= 10% (Siegler & Booth, 2004). PAE was obtained as an index of estimation error on 0−100 number line. Increased PAE indicates increased estimation errors, which also means the low estimation accuracy of individuals. Meanwhile, decreased PAE indicates increased estimation accuracy. For instance, if in the pretest, the mean PAE of all the participants were 22%, I would predict that all the participants were found 22% estimation errors on the 0−100 number line. That is also to say that all the participants have 78% estimation accuracy on the 0−100 number line estimation task.
To calculate each students’ mean PAE, Individual’s PAE = (PAE_1+PAE_2+PAE_3……PAE_10)/N (e.g., because 10 numbers in total were involved in number estimation task, mean PAE is the average of 10 numbers’ PAE). To calculate the group PAE (blocks-group, bundles-group, and control group), the PAE’= (PAE_1^ '+PAE_2^ '+PAE_3^ '……PAE_10^ ')/N (e.g., group mean PAE is the average of mean PAE of each number).
Linearity of mental number line Linearity indexes of 0−100 number line was obtained...
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...stion No.3, the students would scored 1 point for answering “False” or “No” and explain the left or right side of operation was different. For questions No.4 and No.5, they scored 1 point for correct answer (i.e., “1” means 10).
Therefore, each participant could attain 5 points in total at upper level in either pretest or posttest. Overall base 10 score of the participants ranged from 0 to 5 points. None of the students reached the ceiling effect in both pretest and posttest (score 5). Six participants only got “0” score in pretest. None of them got “0” score in posttest. Therefore, all the data from the 58 students were considered valid and applied to the following analysis. The next section presents results for the mathematics instruction data. Across sections, an alpha level of .05 was adopted to indicate statistical significance, unless otherwise noted.
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