Already at a young age, children show large individual differences in math proficiency. Several studies have shown that executive functions play a large role in math acquisition, but most studies treated executive functions as a whole. The aim of this study was to assess the role of the distinct executive functions in the mathematical development of twenty-six normally developing seven-year-old children who were learning addition over ten. Based on previous research, three executive functions were expected to be found in these children: inhibition, shifting and updating. For each executive function, two tasks were administered and a principal component analysis was performed, which confirmed the existence of these three executive functions. The influence of each executive function on mathematical development was measured. Especially updating and shifting were found to be related to mathematical development, while inhibition only played a minor role. These results strengthen the notion that executive functions are important in mathematical development and that the different executive functions have distinct roles in this development. Furthermore, early assessment of  executive functions can help identify children at risk for mathematical difficulties in an early stage.

Aims

Already at a young age, children show large individual differences in mathematical proficiency and around 5-10% even encounter serious difficulties in math acquisition. Although intelligence is related to mathematical performance it only explains a small amount variance and many children with math learning difficulties have an IQ in the normal range.  When investigating the proximal causes of mathematical difficulties, many other studies therefore used a different approach: executive functions as a predictor of mathematical performance. This study also investigated the relationship between executive functions and the development of mathematical skills in children in the first year of primary education. The aim was to distinguish between different executive functions and to investigate the distinct role of each executive function in different aspects of mathematical development.  Unravelling the role of different executive functions in different aspects of mathematical development and an early examination of children’s executive profiles enable predictions of a child’s specific difficulties in mathematics. In addition, it is a step forward in the development of individual intervention programmes which take into account a child’s specific strengths and weaknesses.

Executive functions are the skills that are needed for goal-directed behaviour. Although there is no consensus on the exact number and nature of executive functions, three have been confirmed in previous studies: inhibition, shifting, and updating (Miyake , Friedman, Emerson, Witzki, & Howerter, 2000). Inhibition involves the conscious suppression of a dominant response in favour of another response or no response at all. Shifting involves switching back and forth from one response set to another. Updating accounts for storage of intermediate results and revising this information when necessary. All three executive functions are expected to be involved in mathematics. Inhibition is required when irrelevant information must be suppressed or a false response must be inhibited. Shifting is important when problems require multiple mathematical operations. Updating, finally, is responsible for the storage and retrieval of intermediate results in working memory.

Many previous studies have found a relationship between executive functions and mathematical performance, but most regarded executive functions as a unitary system. In addition, most studies investigated automatised mathematical knowledge.

In this study, all three executive functions were considered separately. The children were tested repeatedly on a mathematical skill that they were learning at the time of the study: addition over ten. Tie problems (e.g. 8 + 8) were chosen because it is known that children learn these relatively quickly.

Methodology

Twenty-six normally developing seven-year-old children were tested on their executive functions and their mathematical performance. Two tasks per executive function were administered. Raven’s Coloured Progressive Matrices were also administered in order to obtain a measure of general intelligence.

Mathematical performance was measured eight times during a four-week period by means of a speeded test containing tie problems over ten. The number of problems solved correctly per minute was taken as a measure of mathematical performance. In the first and last testing session children were also tested on their subtraction knowledge of tie problems (e.g. 16 – 8).

Development has been measured according to Siegler’s (1996) five dimensions of development: source, rate, path, breadth and variability. These five, and the overall error percentage were then related to the three executive functions.

Findings

The principal component analysis confirmed the existence of the three separate executive functions of inhibition, shifting, and updating. Neither function correlated significantly with intelligence. The three factors were related to the five previously mentioned dimensions.

Source

Source concerns the origin of development. In this study the initial level of the children: their performance on the first two measurements, was taken as a measure of source. This indicates how much children had already learned prior to formal instruction. Only updating and shifting correlated significantly with performance on the first two measurements. In a regression analysis only updating was a significant factor, which explained more variance than IQ did, and also explained a significant amount of variance above IQ.

Rate

Rate was measured by subtracting the score of the first two measurements from the score of the last two measurements. Only updating correlated significantly with rate. A regression analysis shows the same results as for source.

Path

Path is related to the sequence of stages children go through. In this study it was investigated whether children showed a leap in performance and whether their improvement fitted a linear curve. No children showed a leap in their performance. They did differ in linearity of improvement, but this was not related to any of the executive functions or to intelligence.

Breadth

This involves generalisibility of acquired knowledge and was tested by means of the subtraction problems. When correlations with individual test sessions are performed, updating and shifting correlated significantly. When the difference between the first and last session was considered, only the correlation with updating was significant.

Error percentage

Error percentages did not correlate significantly with the executive functions, but there was a significant correlation with intelligence. However, error percentage also correlated highly with initial level. When this was controlled for, intelligence but also inhibition correlated significantly with error percentage.

Theoretical and educational significance

The findings show that executive functions are good predictors of mathematical performance, better than intelligence is. They also show that executive functions can be differentiated at age 6/7, and that each executive function has a different influence on mathematical performance. Since early mathematical performance is strongly related to later mathematical skills, this indicates that children weak at executive functioning are at risk of developing later mathematical learning disabilities and extra attention should be paid to those children. The type of problems in mathematics they experience depends on the type of executive weaknesses they have. This knowledge enables the development of early interventions, aiming at both executive function and early mathematical skills, that suit the needs of the individual child.