In the present study, we set out to investigate whether and how central executive load constrains the strategies that children use during arithmetic processing. Using a dual-task paradigm accompanied by the choice/no-choice method, we tested 233 children (115 6^{th} graders, 118 4^{th} graders). Results showed that the impact of central executive load on reaction times and accuracy scores related to strategy use increased with the magnitude of the demands of the central executive, with central executive load playing an important role in strategy use. Sixth graders performed better than 4^{th} graders in the application of appropriate strategies. Children’s adaptability with respect to strategy choice was affected by the type and magnitude of the central executive load; children showed better adaptability under the no-load condition and the inconsistent/low load condition than under conditions with greater load. Grade level affected children’s adaptability with respect to strategy choice, with 6^{th} graders exhibiting significantly better performance than 4^{th} graders. These results confirm that the development of central executive skills contributes to children’s overall strategy use and adaptability. These findings have important implications for understanding the category specificity of central executive working memory in arithmetic cognition and the mechanisms of strategy development in childhood.

Arithmetic estimation, which includes computational estimation, magnitude estimation, and measurement estimation, is an important activity in mathematical cognition. Computational estimation involves a process of approximation in which individuals do not perform numerical calculation, but instead rely on their prior knowledge to provide a rough answer for a given problem. It requires the interaction of mental arithmetic, number concepts, and arithmetic skills (

Children’s estimation strategies have been shown to be affected by arithmetic skills. Some of the most common estimation strategies among Chinese 6^{th} grade children were rounding to omit mantissa (

The most significant age-related changes in children’s arithmetic skills can be characterized in terms of strategy development (^{th} and 6^{th} grades (^{th}, 5^{th} grades and 6^{th} grades time, speed of strategy use changes a lot (^{th} and 6^{th} grades is very important for the development of strategy use. This is also why we included these two grades in our study.

The central executive is the most complex component of working memory (

A perspective grounded in arithmetic strategy not only allows us to better understand the role of various factors and experimental situations, it also allows for a deeper understanding of individual differences in terms of skills, age, and cognitive abilities (^{th}, 5^{th}, and 6^{th} grade children in estimations involving simple single-digit addition, and to explore any pertinent age-related changes. Regardless of load condition, they reported that children most frequently applied a retrieval strategy; age-related differences in strategy execution were not observed under working memory load (

The choice/no choice method provides an unbiased estimate of individuals’ strategy choice (

Based on previous research, the present study aimed to examine children’s strategy performance (strategy selection and execution) in performing an addition estimation task by manipulating the complexity of the central executive load (i.e., load intensity and consistency of the main and secondary tasks). Under the high-load conditions, a successive digit addition task and an alphabetical ordering task were adopted as the secondary tasks; these primarily involve informational operations and updating. Under the low-load conditions, a digital recognition task and a letter recognition task were adopted as the secondary tasks, tasks that primarily involve information encoding and storage. Under the type-consistent conditions, the successive digit addition task and the digit recognition task were adopted as the secondary tasks, as both of these belong to the same category and compete for the same cognitive resources as the main task. Under the type-inconsistent conditions, the alphabetic ordering task and the letter recognition task were adopted as the secondary tasks. Compared with tasks that fall in the same category, tasks belonging to different categories seldom compete for the same resources. By comparing load intensity and load type, we hoped to observe any pertinent differences in the influence of the central executive load on children’s strategy use as a function of increasing age, and to further elucidate the linkage between the central executive and cognitive strategy use. We assumed that under the condition of high level central executive load, both of their performance get worse compared with no load or low level load. And those 4th grade children performed worse.

Arithmetic skills are the ability of complete basic arithmetic. It has an obvious influence on strategy use and strongly influence strategy choice (

A total of 255 children from two ordinary primary schools in China (including 130 4^{th} graders and 125 6^{th} graders) were selected. All participants were required first to complete the arithmetic skills test, and then to simultaneously complete the addition estimation task and the secondary task (if any). Based on this testing, 233 subjects with normal eyesight or corrected normal eyesight were retained in the final sample (113 boys and 120 girls; 118 4^{th} graders and 115 6^{th} graders; average age, 10.63 ± 1.27 years).

In this study, “consistent” means both the main task and the secondary task involve digital operations. Correspondingly, “inconsistent” means two tasks involve different operations (main task involving numbers, secondary task involving letters). A 5 (load situation: consistent/high load, inconsistent/high load, consistent/low load, inconsistent/low load, and no load) × 3 (strategy use condition: best choice (C1), rounding up (C2), and rounding down (C3)) × 2 (grade level: 4 and 6) mixed experimental design was used. Load and grade level were implemented as between-subjects variables (the participants in each group are shown in Table

Numbers of participants allocated in different load situations.

Consistent–high load | Consistent– low load | Inconsistent– high load | Inconsistent–low load | No load | Total | |
---|---|---|---|---|---|---|

6^{th} grade |
22 | 19 | 25 | 23 | 26 | 115 |

4^{th} grade |
25 | 23 | 20 | 25 | 25 | 118 |

Total | 47 | 42 | 45 | 48 | 51 | 233 |

The French Kit test was adopted (

Thirty two-digit addition estimation problems (for example, 76 + 42) were used in the main task, including 15 rounding-down problems (in which the rounding-down strategy was required for estimation, such that 51 + 78 becomes 50 + 70 = 120), 15 rounding-up problems (in which the rounding-up strategy was required for estimation; thus, 74 + 69 becomes 80 + 70 = 150). In half of the problems, the unit digit of the first addend was less than 5, and the unit digit of the second addend was greater than 5. The other half of the problems was structured in the opposite manner. In half of the problems, the greater addend was on the left (e.g., 86 + 52), and in the remaining half of the problems, the greater addend was on the right (e.g., 43 + 86). In addition to the above constraints (

Materials adapted from Han and Kim were used (

The experiment was divided into three parts: the best-choice condition (C1), under which participants were instructed to choose between two given strategies (rounding up and rounding down) to arrive at an answer that approximated the accurate sum; the no-choice/rounding-up condition (C2), under which participants were instructed to use only the rounding-up strategy to arrive at their estimates; and the no-choice/rounding-down condition (C3), under which subjects were instructed to apply only the rounding-down strategy to estimate the sum. Participants were instructed to type in their responses as quickly and accurately as possible.

The rounding-up strategy means rounding the two addends up to their nearest tens (73 + 49 → 80 + 50, the answer is 130); The rounding-down strategy involved adjusting both addends down to their nearest tens (73 + 49 → 70 + 40, the answer is 110). A mixed strategy in which one addend was rounded up and the other was rounded down was not permitted throughout the entire experiment.

To avoid any influence of the no-choice conditions on the execution of strategies in the choice condition, we first tested all participants with stimuli from C1 (the best-choice condition), followed by C2 and C3. The interval between any two conditions was 5 minutes, each of 30 trials.

Each of the participants completed 10 practice trials to become familiarized with the experimental procedure and tasks before the formal experiment.

In no-load condition, participants were only required to complete the estimation task (Figure

Flow chart of no-load condition.

Flow chart of consistent/high-load condition.

Flow chart of consistent/low-load condition.

Flow chart of inconsistent/high-load condition.

Flow chart of inconsistent/low-load condition.

Reaction times and accuracy scores obtained under the no-choice rounding-up condition provide an unbiased estimate of the execution of the rounding-up strategy; similarly, the execution of the rounding-down strategy is reflected by these measures under the no-choice rounding-down condition. Reaction times for responses in which participants failed to apply the given strategy were excluded.

Using the arithmetic skill score as a covariate and reaction time for the estimation task in C2 and C3 as the dependent variable, we conducted a 2 (strategy use condition) × 5 (load situation) × 2 (grade level) repeated-measures analysis of variance. Results revealed a significant main effect of arithmetic skill (_{(1,222)} = 38.11, ^{2}_{(1,222)} = 33.71, ^{2}_{(4,222)} = 10.38, ^{2}_{(1,222)} = 46.03, ^{2}^{th} graders were faster than those of 4^{th} graders.

A noteworthy interaction emerged between strategy use condition and load situation (_{(4,222)} = 4.72, ^{2}

Reaction time of strategy execution in different strategy use conditions and load situations. C–H: consistent-high load; C–L: consistent-low load; IC–H: inconsistent-high load; IC–L; inconsistent-low load; NO: no load.

As can be seen in Figure _{(4,222)} = 2.75, ^{2}^{th} graders, there were no significant differences between any of the following: the consistent/high-load and inconsistent/high-load conditions (^{th} graders, strategy execution time was affected by load intensity. However, low load had no effect on strategy execution time; it appears that it is only when the central executive system is sufficiently taxed that an impact on 6^{th} graders’ strategy execution time was observed, slowing down response times. In contrast to load intensity, the influence of load type on strategy execution time at this grade level was not substantial. The inconsistent/high load condition seems to lead to detrimental effects in young children. Similar to the results before, for younger children, alphabetical ordering is worse than for 6^{th} graders. Therefore, they are more likely to be affected.

Reaction time of strategy execution in different grades and load situations. C–H: consistent-high load; C–L: consistent-low load; IC–H: inconsistent-high load; IC–L; inconsistent-low load; NO: no load.

A different pattern emerged for 4^{th} graders. The difference between the consistent/high-load and inconsistent/high-load conditions was not significant (^{th} graders, whereas low load had little effect on 6^{th} graders.

No significant interaction was found between strategy use and grade level (_{(4,222)} = 1.86, ^{2}^{th} graders under both the rounding-up and rounding-down strategy conditions. The three-way interaction of strategy use condition by grade level by load situation was not significant (_{(4,222)} = 1.34, ^{2}

Using participants’ accuracy scores for the estimation task under conditions C2 and C3 as the dependent variable and arithmetic skill as the covariate, we conducted a 2 (strategy use condition) × 5 (load situation) × 2 (grade level) repeated-measures analysis of variance. The results yielded no significant main effect of arithmetic skill (_{(1,222)} = 0.24, ^{2}_{(1,222)} = 1.95, ^{2}_{(1,222)} = 12.76, ^{2}^{th} graders exhibiting better accuracy than 4^{th} graders. There was also a main effect of the load situation (_{(4,222)} = 5.99, ^{2}

There was a significant interaction between strategy use condition and grade level (_{(4,222)} = 6.82, ^{2}^{th} (^{th} graders (^{th} (^{th} graders (^{th} and 6^{th} grades appears to be an important period in terms of changes in performance accuracy related to strategy execution. There were no significant interactions between strategy use condition and load situation (_{(4,222)} = 1.02, ^{2}_{(4,222)} = 0.99, ^{2}_{(4,222)} = 0.93, ^{2}

Reaction time and accuracy scores of subjects under the best-choice condition (C1) reflect their strategy choices. We excluded reaction times for trials in which participants failed to apply one of the two targeted strategies (rounding up or rounding down). The accuracy score for each rounding-up condition was computed as the number of trials in which participants correctly applied the rounding-up strategy when it was optimal to do so divided by the total number of trials in which the rounding-up strategy was used. Similarly, the accuracy score for each rounding-down condition equaled the number of trials in which participants correctly applied the rounding-down strategy when it was optimal to do so divided by the total number of trials in which the rounding-down strategy was used.

Using participants’ reaction times for executing rounding-up and rounding-down strategies under the best-choice condition (C1) as the dependent variable and arithmetic skills as the covariate, we conducted a 2 (strategy type) × 5 (load situation) × 2 (grade level) repeated measures analysis of variance. The results revealed a robust main effect of arithmetic skill, _{(1,222)} = 16.73, ^{2}_{(1,222)} = 5.32, ^{2}_{(4,222)} = 3.52, ^{2}_{(1,222)} = 3.50, ^{2}_{(4,222)} = 1.92, ^{2}_{(1,222)} = 0.30, ^{2}_{(4,222)} = 0.69, ^{2}_{(4,222)} = 0.31, ^{2}

We conducted a 2 (strategy type) × 5 (load situation) × 2 (grade level) repeated-measures analysis of variance in which the dependent variable was accuracy scores for rounding-up and rounding-down strategies on the estimation task under the best-choice condition (C1), and arithmetic skill was the covariate. The results revealed no significant main effect of arithmetic skill (_{(1,222)} = 0.57, ^{2}_{(4,222)} = 3.29, ^{2}_{(1,222)} = 14.18, ^{2}^{th} graders was significantly higher than that of the 4^{th} graders. However, the main effect of strategy type was very weak (_{(1,222)} = 0.28, ^{2}_{(4,222)} = 1.97, ^{2}_{(1,222)} = 0.04, ^{2}_{(4,222)} = 2.08, ^{2}_{(4,222)} = 0.49, ^{2}

We defined the adaptability of strategy choice in terms of the ability to choose the strategy that most closely approximated the accurate answer to the addition problem (_{(1,222)} = 0.27, ^{2}_{(4,222)} = 4.14, ^{2}_{(1,222)} = 13.41, ^{2}^{th} graders was considerably higher than that of 4^{th} graders, implying that the period between the 4^{th} and 6^{th} grades may be a developmentally important period with regard to this particular skill. However, there was no significant interaction of grade level by load situation (_{(4,222)} = 2.20, ^{2}^{th} graders was lower than that among 6^{th} graders under the various load situations. The adaptability of children’s strategy choices displayed a relatively consistent trend across all five load situations (see Figure

Adaptability of strategy choice in 4^{th} and sixth- grade children under different load situations. C–H: consistent-high load; C–L: consistent-low load; IC–H: inconsistent-high load; IC–L; inconsistent-low load; NO: no load.

This study examined the impacts of various loads on central executive functioning in children’s estimation strategies at different ages. Results showed that the central executive load affected children’s strategy performance. The heavier the load is, the greater the impact on children and 4^{th} grade children were more susceptible.

The presence of any central executive load resulted in interference, as seen in the overall decline of strategy execution efficiency (reaction times and accuracy scores). To some degree, this is consistent with Wang and Chen’s study (

The specific pattern of these effects appears to change with age. Fourth grade children were more sensitive to slight increases in central executive load due to their limited working memory resources, with the result that their performance on the main task deteriorated relative to the single task (no-load condition) even under minimal load. When working memory resources are not already subject to heavy strain, the competition between the category-consistent primary and secondary tasks is considerably higher than the competition that arises in category-inconsistent situations. Accordingly, we found that 6^{th} grade children were able to complete dual tasks with ease when the secondary task was inconsistent, creating less competition for resources.

Si, Yang, Jia, and Zhou (

Increasing age resulted in a gradual improvement in the speed and accuracy of children’s strategy execution under conditions of central executive load. Both reaction times and accuracy scores for strategy execution showed considerably stronger performance among 6^{th} graders than among 4^{th} graders. The development of working memory resources and executive functions may play an important role in age-related differences in strategy use (

Our study further confirms that the influence of central executive load on strategy execution changes with age. Strategy execution among 6^{th} grade children is superior relative to that that among 4^{th} graders, and accuracy improves for the more complex rounding-up strategy. This supports the notion that the period between grades 4 and 6 represents a key developmental period with respect to strategy execution in estimation tasks (

The presence of the central executive load of any type or intensity affected children’s reaction times and accuracy scores when strategy choice was involved. Strategy use involves processes related to both selection and execution. Of the two processes, strategy choice requires greater central executive capacities and is more easily affected by the central executive load than strategy execution. However, Imbo and Vandierendonck’s (^{th} graders increased relative to that of 4^{th} graders, with the older children choosing and executing more effective strategies, including the more complex rounding-up strategy, to solve problems. This indicates that changes in children’s strategy development are influenced by age and by changes in the central executive function (^{th} grade, which also might influence these results.

We observed effects of various types and intensities of load on the adaptability of children’s strategy choice. In situations with no load or inconsistent/low load, children exhibited greater accuracy and better adaptability in their strategy choices. Previous findings have shown that the presence of any load affected the adaptability of adults’ strategy choices (^{th} grade children. However, children’s adaptability with respect to strategy choice remains in need of further development, as evidenced by previous findings (

Our results indicate that among 4^{th} graders, the presence of any degree of central executive load affected strategy use. By the 6^{th} grade, the impact of low degrees of central executive load was considerably weaker, reflecting the effects of increasing age and stronger executive function. Overall, the central executive load had a greater impact on children’s strategy choice than on their strategy execution. We can conclude that the complexity of the central executive load not only increases the intensity of resource competition, but also affects the development of strategy execution and strategy choice in childhood.

The authors have no competing interests to declare.

Project 31371048 supported by National Natural Science Foundation of China, project ZR2010CM059 supported by Shandong Province Natural Science Foundation, and the Key Subject Funds of Shandong Province, P. R. China (2011–2015).