Performance in terms of speed and accuracy should improve with age [ 4 , 5 ]. Moreover, as mentioned earlier Ganor-Stern and Siegler [ 9 ] reported that in the estimation production task, about half of the estimates produced by 6 th graders were ten times smaller than the exact answer.
With age there should be more variability in strategy use and more adaptivity in strategy choice [ 21 , 23 ]. Furthermore, with respect to the frequency of use of the two main strategies found in this task—the approximate calculation and the sense of magnitude, the following two possibilities were raised.
Due to the training in multiplication and in rounding procedures given in school and the emphasis given to solving mathematical problems using calculation, children might be more inclined to use the approximate calculation strategy than adults, who in turn might rely more on the sense of magnitude strategy at least for far reference numbers. Ninety-two participants took part in this study, thirty-two 4 th graders, thirty 6 th graders, and thirty college students.
Four 4 th graders, two 6 th graders and two college students were excluded from the analysis due to chance level performance or an exceptionally long response time, leaving 28 participants in each age group. The mean age was The children were from two public elementary schools in the south of Israel. Two research assistants administered the experiment in a quiet room in school or in the laboratory.
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The procedure was approved by the ethics committees of the Israeli Ministry of Education and of Achva Academic College. The college students provided written informed consent to participate in this study. Adhering to the policy of the Ministry of Education IRB, the parents of the school children denied consent by returning an enclosed form.
The experiment was conducted on a Lenovo Thinkpad laptop computer with a inch screen. The software was programmed in E-Prime [ 24 ].
The stimuli set was composed of forty 2D x 2D multiplication problems. Following Ganor-Stern [ 12 ], the problem set was constructed with the following restrictions. There were no tie problems. No operand had 0 as units digit. No reversed orders of operands were used 53 x 76 was not used with 76 x The operands ranged between 13 and 95, and the exact answers were in the range of — In half of the problems the larger operand was on the right, while in the other half the larger operand was on the left.
Each problem was associated with 4 reference numbers: one which was about one fifth of the exact answer, one which was about one half of the exact answer, one which was about twice the exact answer, and one which was about 5 times the exact answer.
Reference numbers were rounded to the nearest hundred. The problems were arranged in four lists that were counterbalanced across participants. Thus, each participant responded to only one list. Within each list, each problem appeared once; across lists, each problem appeared with each of the four reference numbers.
Within each list, in half of the trials the exact answer was larger than the reference number, and in the other half it was smaller than the reference number. The experiment was conducted individually. The participant sat about 50 cm from the computer screen. In each trial a multiplication problem appeared on the screen with a reference number below it see Fig 1.
Participants were asked to estimate whether the answer for each problem was larger or smaller than the reference number. They had to press the "L" key if they estimated it to be larger than the reference number, and the "A" key if they estimated it to be smaller.
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The participants were given two examples of the task, together with the corresponding correct responses, to make sure that they understood the task requirements. Participants were explicitly told that they should not solve the problems exactly, but should only estimate whether the answer was larger or smaller than the given number. The order of trials was random. Participants were not allowed to use calculators or paper and pencils for calculation.
Participants did not receive any feedback for their responses. In the first 8 trials, participants responded by pressing the computer keys only, while in the remaining 32 trials, after they pressed the response key for each trial, they were asked to describe how they reached their decision.
The experimenter documented their descriptions. The results section includes analyses of accuracy, speed, and strategy use. Responses that took longer than 2.
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As can be seen in Fig 2 top panel , PE was lower when the exact answer was far from the reference number PE was also lower when the exact answer was larger than the reference number PE was It was Similar patterns appeared in the speed analysis Fig 2 , bottom panel. RT was shorter when the exact answer was far 8. RT was shorter when the exact answer was larger than the reference number 8. Exactly the same number of 6 th graders employed one strategy consistently, however the distribution of strategy use differed with 10 participants implementing the approximate calculation strategy and 11 using the sense of magnitude strategy.
Thus, both the group level information and the individual level analysis showed a clear increase in the use of the approximate calculation strategy with age. Since most participants in all age groups used the two strategies to solve the task, the next step was to investigate how participants decided which strategy to use for which item. Thus, the frequency of use of each strategy was entered into an analysis of variance with age, strategy, size of the reference number relative to the exact answer, and distance as independent variables.
This analysis was limited to the participants that used both strategies 19 4 th graders, 18 6 th graders, and 22 adults. The two effects involving the strategy variable are theoretically important see Fig 3. As mentioned in the Introduction, the two strategies are expected to produce different patterns of speed and accuracy, with the approximate calculation strategy expected to produce accurate but slow responses, and the sense of magnitude strategy expected to produce fast and less accurate responses [ 12 , 13 ].
Moreover, the advantage of the approximate calculation in accuracy is supposed to be especially pronounced for close reference numbers. To examine the effects of age and strategy use on accuracy and speed two analyses were conducted. The first looked at the accuracy and speed of participants who used a single strategy throughout the whole experiment.
Among the 4 th graders, 3 used the approximate calculation strategy exclusively and 6 the sense of magnitude strategy solely, among the 6 th graders, 6 used only the approximate calculation strategy and 4 solely the sense of magnitude strategy, and among the adults, 5 used only the approximate calculation strategy and one participant used only the sense of magnitude strategy.
Since there was only one adult that used the sense of magnitude exclusively the following analysis was limited to the children groups. The RT analysis Fig 4 bottom panel has shown that as expected, RT was shorter for the sense of magnitude strategy 6. Importantly, the improvement in speed with age was larger for the approximate calculation strategy The interaction between strategy and distance.
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The second analysis was conducted on participants that used both strategies. It included age as a between-participant variable and strategy as a within-participant variable. Since there were not enough trials per strategy the data were collapsed across distance and size of the reference number. There were no significant effects in the analysis of PE. Despite the frequent use of the computational estimation skill in everyday life, relatively little is known about how people solve such tasks, and how this skill develops with age. The present research is the first to examine developmental patterns in the computational estimation comparison task by looking at the accuracy, speed, and strategy use of 4 th graders, 6 th graders, and college students.
The developmental pattern between children and adults was reflected in three measures. First, there was an increase in accuracy with age. Importantly, this increase was limited to trials when the reference number was close to the exact answer. Second, speed improved with age across conditions. Third, the dominant strategy changed with age.
In 4 th grade the sense of magnitude was the most common strategy, in 6 th grade the two strategies were used equally often, while in the adults group the approximate calculation strategy was the most common.
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The transition from frequent use of the sense of magnitude strategy into using mainly the approximate calculation strategy with age can explain why the advantage in accuracy for the adults group was limited to the close reference numbers. The faster and less attention demanding sense of magnitude strategy is likely to produce a correct response when the reference number is far, but not when it is close.
The preference of fourth-graders to use the sense of magnitude strategy is probably due to the fact that they do not fully master the calculation skills at this stage, and thus cannot use the approximate calculation strategy effectively.