# The Effect of Central Executive Load on Fourth and Sixth Graders’ Use of Arithmetic Strategies

## Abstract

##### DOI: http://doi.org/10.5334/pb.360
Accepted on 14 Jun 2017            Submitted on 25 Jul 2016

## Introduction

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 (Si, 2002). Examining the mechanisms underlying computational estimation and its development may deepen our understanding of mathematical and general problem solving ability (Si, Yang, Jia, & Zhou, 2012). A substantial number of studies have shown that, when solving arithmetical problems, there were varieties of strategies which children choose to use (Lemaire & Lecacheur, 2002; Si & Zhang, 2003; Chen & Geng, 2005; Lemaire & Callies, 2009). Mental arithmetic refers to there is no external tools (such as a pen, calculator, etc.) in the process of arithmetic operations activities (Campbell, 2005). Arithmetic skills are, however, the individual ability of complete basic arithmetic (Imbo, 2007). These are two different concepts.

Children’s estimation strategies have been shown to be affected by arithmetic skills. Some of the most common estimation strategies among Chinese 6th grade children were rounding to omit mantissa (Si & Zhang, 2003). Rounding has been investigated in many previous studies (Si, Yang, Jia, & Zhou, 2012; Lemaire, Arnaud & Lecacheur, 2004; Imbo, Duverne, & Lemaire, 2007; Lemaire & Lecacheur, 2010). In addition problems, rounding includes two different variants: rounding down (in which both of two operands are rounded down to the nearest whole tens; thus, 28 + 63 becomes 20 + 60 = 80) and rounding up (in which both of two operands are rounded up to the nearest whole tens; thus, 28 + 63 becomes 30 + 70 = 100). In the present study, these two strategies will be used to investigate children’s ability to solve arithmetic problems effectively and flexibly.

The most significant age-related changes in children’s arithmetic skills can be characterized in terms of strategy development (Lemaire, 2010). Changes in strategy choice may be affected by age-related changes pertaining to the central executive component of the working memory system (Lemaire, 2010; Lemaire & Lecacheur, 2011). However, to date, it remains unclear how age-related changes in the central executive system constrain the development of children’s arithmetic strategy use. Research has shown those children’s strategy use changes rapidly in the period between the 4th and 6th grades (Lemaire, Lecacheur, & Farioli, 2000). During the 4th, 5th grades and 6th grades time, speed of strategy use changes a lot (Lemaire, 2010). That is to say, the time between 4th and 6th 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 (Baddeley, 2010). It is an attention control system and its involvement is necessary for us to perform numerous higher-level cognitive activities such as chess playing. The central executive system can be divided into four distinct executive functions: memory updating, inhibition, switching, and dual-task coordination (Baddeley, 1996; Collette & Linden, 2002). Existing data have shown that various executive functions (memory updating, inhibition, switching, dual-task coordination) can affect children’s arithmetic performance and strategy use (Bull & Scerif, 2001). Researchers found links among dual-task coordination, switching, inhibition, updating, children’s strategy selection and execution (Chen & Wang, 2009). Although the various executive functions are separable, they are not completely independent; they have a common base in central executive working memory. Central executive working memory plays an important role in adults’ arithmetic behaviors (Baddeley, 1996). The central executive may affect strategy execution and strategy choice by participating in the carry operation involved in the process of calculation (Caviola, Mammarella, Cornoldi, & Lucangeli, 2012; Imbo, Vandierendonck, & Vergauwe, 2007). Most prior research has relied on measures of working memory span to explore the role of central executive load on the use of arithmetic cognitive strategies. For example, British children with low working memory spans were not able to select an efficient strategy when performing a multiplication task (Steel & Funnellf, 2001). In the latter study, retrieval was the fastest and least error-prone strategy, counting-in-series was the slowest and most error prone, and they also found that children mainly used mixed strategies. Working memory span affected children’s strategy distribution and implementation (Chen & Wang, 2009).

The choice/no choice method provides an unbiased estimate of individuals’ strategy choice (Siegler & Lemaire, 1997; Luwel, Onghena, Torbeyns, Schillemans, & Verschaffel, 2009). It involves two types of experimental conditions: the choice condition, under which subjects may freely choose which strategies they are going to use to solve problems, and the no-choice condition, under which subjects must use the specified strategy to solve all problems. The number of no-choice conditions should be equal to the number of possible strategies in the choice condition. As in our previous work (Si, Yang, Jia, & Zhou, 2012; Sun, Si, & Xu, 2012), the present study also implemented a best-choice condition, under which subjects were instructed to choose the most appropriate strategy to solve the given problem; this was done to further reveal the degree of flexibility of individuals’ strategy use. To prevent a general carry-over effect from the no-choice condition, the choice conditions were presented first. Additionally, the present research combined a dual-task paradigm with the choice/no choice method to examine children’s strategy use in a computational estimation task under different conditions of central executive load. The dual-task paradigm has been frequently implemented in both adult and child studies (Imbo & Vandierendonck, 2007; DeStefano & LeFevre, 2004).

Arithmetic skills are the ability of complete basic arithmetic. It has an obvious influence on strategy use and strongly influence strategy choice (Thevenot, Fanget, & Fayol, 2007; Imbo, Vandierendonck & Rosseel, 2007). A number of researchers have used arithmetic skill as a covariate to examine individuals’ strategy use (Imbo & LeFevre, 2010; Barrouillet & Lépine, 2005; Miyake, Friedman, Emerson, Witzki, Howerter, & Wager, 2000). In the present study, we controlled for arithmetic skill and examined whether the central executive working memory load has a separate influence on children’s strategy use over time, independent of arithmetic skill.

## Method

### Participants

A total of 255 children from two ordinary primary schools in China (including 130 4th graders and 125 6th 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 4th graders and 115 6th graders; average age, 10.63 ± 1.27 years).

### Design

Table 1

Numbers of participants allocated in different load situations.

6th grade 22 19 25 23 26 115
4th grade 25 23 20 25 25 118
Total 47 42 45 48 51 233

### Materials

#### Arithmetic tests

The French Kit test was adopted (French, Ekstrom, & Price, 1963). The test contains two subtests; one involves complex addition with three addends, and the other including mixed subtraction and multiplication problems involving two subtrahends or two multipliers. Each subtest is divided into two parts; each part contains 60 questions with a 2-minute answer time. The number of correct responses corresponds to the test score. Many previous studies have adopted this tool to measure individuals’ arithmetic skills (Thevenot, Fanget, & Fayol, 2007; Ai, Zhang, Si, Lu, & Zhang, 2016; Huang, Feng, Si, Zhang, & Wang, 2016).

### Procedure

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.

Figure 1

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## Results

### Strategy execution

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.

#### Reaction time

Figure 6

Figure 7

No significant interaction was found between strategy use and grade level (F(4,222) = 1.86, p > 0.05, η2 = 0.008). Sixth graders showed similar advantages in terms of speed of responding relative to 4th 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 (F(4,222) = 1.34, p > 0.05, η2 = 0.023).

#### Accuracy

There was a significant interaction between strategy use condition and grade level (F(4,222) = 6.82, p < 0.05, η2 = 0.030). A significant difference emerged between the 4th (M = 0.869, SD = 0.012) and 6th graders (M = 0.967, SD = 0.012) under the no-choice and rounding-up conditions. However, the no-choice and rounding-down conditions yielded no significant differences between 4th (M = 0.945, SD = 0.007) and 6th graders (M = 0.936, SD = 0.007). These results indicate that strategy complexity affects children’s estimation accuracy. The period between the 4th and 6th 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 (F(4,222) = 1.02, p > 0.05, η2 = 0.018), between grade level and load situation (F(4,222) = 0.99, p > 0.05, η2 = 0.018), or among strategy use condition, grade level, and load situation (F(4,222) = 0.93, p > 0.05, η2 = 0.016).

### Strategy choice

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.

Figure 8

## Discussion

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 4th grade children were more susceptible.

### Effects of central executive load on children’s strategy execution at different ages

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 6th graders than among 4th graders. The development of working memory resources and executive functions may play an important role in age-related differences in strategy use (Lemaire & Lecacheur, 2011; Hodzik & Lemaire, 2011). Working memory capacity develops with age. It exhibits particularly rapid development before 8 years of age, slower development between 8 and 20 years of age, and then begins to decline after the age of 20 (Li, Bai, & Shen, 2006).

Our study further confirms that the influence of central executive load on strategy execution changes with age. Strategy execution among 6th grade children is superior relative to that that among 4th 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 (Lemaire, Lecacheur, & Farioli, 2000; Lemaire & Lecacheur, 2002). The results indicate that increasing age and experience are associated with greater accuracy and faster reaction times among Chinese children performing two-digit addition estimation tasks and that the execution of rounding-up and rounding-down strategies shows a development progression, with particularly strong developmental effects observed for the rounding-down strategy.