It is well established that learning mathematics presents a challenge for many children and young people. This can lead to negative consequences for job prospects and quality of life (Parsons and Bynner 2005). Despite educational initiatives to improve mathematics achievement there has been a disappointing lack of improvement in mathematics outcomes in many Western societies (Vorderman et al. 2011). Given this lack of progress, researchers have attempted to better understand the cognitive processes that underlie mathematics performance. An improved theoretical understanding of the factors involved in mathematics processing can provide the starting point from which to develop pedagogy to support mathematics learning in all young people.
Over the past few decades, researchers have identified two classes of cognitive skills that are important for mathematics achievement. The first of these concerns domain-specific skills such as symbol knowledge, counting skill, and underlying numerical representations. Alongside these, researchers have identified domain-general skills which are involved in learning in many areas but which are particularly important for mathematics (e.g. language, IQ and spatial ability). Particular attention has been paid to executive functions—the skills required to monitor and control thought and action—and the role they play in learning and performing mathematics (see reviews by Cragg and Gilmore 2014; Bull and Lee 2014). Three types of executive functions have been identified: monitoring and manipulating information in mind (working memory), suppressing distracting information and unwanted responses (inhibition), and flexible thinking (shifting). To date, few models of mathematical cognition have considered the role of executive function skills, particularly inhibition. LeFevre et al. (2010) identified a role for attentional processes in their Pathways Model. This model proposed that attentional skills have a direct impact on mathematical performance in a variety of domains independent of linguistic or quantitative skills. However, the specific role of inhibition skill was not specified in this model.
There is a wealth of evidence that working memory is involved in mathematics (see reviews by DeStefano and LeFevre 2004; Raghubar et al. 2010). More recently, the focus has turned to inhibition and it has been hypothesized that individuals with higher levels of inhibitory control are more successful in mathematics. In the sections below, we first review the existing evidence for a link between inhibition and mathematics before discussing different types of inhibition task and different components of arithmetic. Finally, we introduce the present study.
Inhibition and mathematics performance
In the past 5 years there has been a steady increase in the number of studies that have explored the role of inhibition skills in mathematics performance. The majority of studies have employed correlational methods to explore the relationship between performance on tests of inhibition and concurrent mathematics achievement. For example, children’s performance on experimental inhibition tasks are related to their school mathematics grades (Brock et al. 2009; Visu-Petra et al. 2011) as well as performance on standardized mathematics tests (Nayfield et al. 2013; St Clair-Thompson and Gathercole 2006). A smaller number of studies have found that inhibition predicts future success in mathematics (Blair and Razza 2007; Clark et al. 2010; Swanson 2011). Converging evidence for a link between inhibition and mathematics comes from studies that have compared the inhibition skills of different groups of children. Studies by Szucs et al. (2013), Wang et al. (2012) and Winegar (2013) found that children with identified mathematical learning difficulties performed more poorly on inhibition tasks than children with average performance in mathematics.
In contrast to these findings, a number of studies have failed to find evidence for a link between inhibition skills and mathematics: Waber et al. (2006) found weak relationships between experimental measures of inhibition and curriculum measures of mathematics performance; Miller et al. (2013) found that inhibition skills were not a unique predictor of mathematics performance and Monette et al. (2011) found that inhibition skills predicted future reading/writing achievement but not future mathematics achievement. There is some evidence that inhibition skills may only be related to mathematics outcomes if shifting skills are not taken into account. Both Bull and Scerif (2001) and Van der Ven et al. (2012) found that measures of inhibition were no longer related to mathematics once shifting skills were included in the model. However, Espy et al. (2004) found that inhibition did predict mathematics even after controlling for both working memory and shifting skills.
To summarize, there is mixed evidence concerning the relationship between inhibition and mathematics performance. Although a number of studies have found a positive relationship, this appears to be more nuanced than originally proposed. Previous studies have been conducted with participants across a wide range of ages. This is an important consideration as inhibition skills mature and the nature of mathematics content changes with age, thus some of the inconsistencies outlined above may reflect changes in the role of inhibition skills across age. Two further factors may be important in explaining these inconsistencies: the type of inhibition task and the nature of the mathematics test. These will be considered in further detail below.
Types of inhibition task
Previous studies have employed a wide range of tasks to assess inhibition. These different tasks tap into varying aspects of inhibition skill. A distinction is commonly made between response inhibition and interference control (e.g. Nigg 2000). Interference control concerns the suppression of distracting information, either internal or external, which leads to an alternative non-desired response. The Stroop task is the best-known measure of interference control. In this task participants are required to focus on and respond to one aspect of a given stimulus (e.g. the colour of ink a word is written in) whilst ignoring other features of the stimulus (e.g. the word itself). Stroop tasks have frequently been employed in studies exploring the relationship of inhibition with mathematics performance (e.g. Bull and Scerif 2001; Lemaire and Lecacheur 2011; Monette et al. 2011; Navarro et al. 2011; St Clair-Thompson and Gathercole 2006; Szucs et al. 2013; Van der Ven et al. 2012; Visu-Petra et al. 2011).
A second form of inhibition is response inhibition. This concerns the suppression of a prepotent motor response and is often measured using Go/No-Go or Stop-signal tasks. In these tasks participants are required to frequently make one type of response unless they receive a signal to withhold the response. Inhibitory control performance is indexed by a failure to withhold the prepotent response. A smaller number of studies have used these types of tasks when exploring the relationship with mathematics achievement (De Weerdt et al. 2013; Monette et al. 2011; St Clair-Thompson and Gathercole 2006). At present there is evidence from different studies to suggest that both interference control and response inhibition are related to mathematics achievement (St Clair-Thompson and Gathercole 2006; Szucs et al. 2013); however, the relative importance of each type of inhibition remains unclear. Moreover, the mechanisms by which they support mathematics performance are likely to differ.
A further distinction that applies to both interference control and response inhibition tasks is between tasks that involve the inhibition of domain-relevant and domain-irrelevant information. It has been proposed that, rather than a single inhibitory system that is applied across all domains, there are multiple, domain-specific, inhibitory control systems (Egner 2008). Consequently, participants may show differing levels of inhibitory control according to the nature of the information they are being required to inhibit. More importantly here, mathematics achievement may be more strongly related to the inhibition of numerical information, rather than applying to inhibition skills more generally (Bull and Scerif 2001). To explore this question, multiple versions of inhibition tasks have been employed which involve the inhibition of either numerical or non-numerical information. For example, alongside the standard colour-word stroop, which involves non-numerical information, studies have employed stroop tasks that involve the inhibition of numerically relevant information. For example Szucs et al. (2013), Wang et al. (2012), Zhang and Wu (2011), and Navarro et al. (2011) used a number-size stroop task in which participants are required to select the numerically highest digit whilst ignoring the size of the digits on the screen (e.g. 3 vs. 5). Bull and Scerif (2001) and Wang et al. (2012) have also made use of a number–quantity stroop task in which participants are required to name how many items are in a set while ignoring the digit itself (i.e. to respond “three” to the stimulus 555). Similarly, numerical and non-numerical versions of Go/No-Go tasks (De Weerdt et al. 2013) and random generation tasks (Winegar 2013) have also been developed. These alternative task versions have been used to explore the hypothesis that the relationship between inhibition and mathematics achievement is specific to the inhibition of numerically relevant information, with mixed results.
In favour of the domain-specific inhibition hypothesis, both Bull and Scerif (2001) and Navarro et al. (2011) found that only performance on the number–quantity version of the stroop task and not performance on the colour–word version correlated with mathematics achievement. Similarly, when comparing the performance of children with and without mathematics learning difficulties, both Szucs et al. (2013) and Wang et al. (2012) found that group differences were only significant for numerical stroop tasks rather than non-numerical versions. However, other studies have failed to find domain-specific effects with stroop tasks (Zhang and Wu 2011) or Go/No-Go tasks (De Weerdt et al. 2013).
In summary, there is some evidence to suggest that the content of inhibition tasks has an impact on the relationship with mathematics achievement. It is unclear, however, if this effect only arises in tasks involving the processing of Arabic digits or, alternatively, whether this holds for numerically relevant information more generally. Both the number–size and number–quantity versions of the stroop task, for which domain-specific effects have been observed, involve Arabic digits. Bull and Scerif (2001) suggest that, before conclusions about domain-specific numerical inhibition effects can be justified, versions of the stroop task with different types of numerical stimuli should be explored. An alternative task, which involves numerically relevant information but does not include Arabic digits, is the dot comparison task. In this task, which was originally developed to measure numerical magnitude processing, participants are shown pairs of dot arrays and are asked to select the more numerous array while ignoring the visual characteristics (e.g. dot size, density, area) of the arrays. Typically two types of trials are included: congruent trials, in which the more numerous array also has the larger visual characteristics, and incongruent trials, in which the less numerous array has larger visual characteristics. It has been proposed that solving incongruent trials of this task therefore has significant inhibitory control demands (Fuhs and McNeil 2013; Gilmore et al. 2013; Nys and Content 2012) and that the difference in performance on congruent and incongruent trials therefore provides a measure of inhibition skills. In line with this, Szucs and colleagues (2013) found that children with mathematics learning difficulties had larger congruency effects on a dot comparison task than controls.
Congruency effects on a dot comparison task therefore provide a measure of inhibition in a numerical context without the use of Arabic digits themselves. Performance on this task can be contrasted with an equivalent task involving non-numerical information. A suitable task here is the animal-size stroop task that has been used by Szucs et al. (2013). In this task participants are shown two animal images and are asked to select the animal that is larger in real life, while ignoring the size of the images on the screen. Again, participants see both congruent trials, in which the animal that is larger in real life is also larger on screen, and incongruent trials, in which the animal that is smaller in real life is larger on screen. Performance on incongruent trials of both the dot comparison and animal stroop tasks involve ignoring the superficial task-irrelevant visual characteristics of the stimuli, but the tasks differ in whether the relevant processing is numerical or not. These tasks therefore allow us to test whether both domain-specific (i.e. numerical) and domain-general (i.e. non-numerical) measures of interference control are related to mathematics achievement.
Components of arithmetic
A second factor that may help to explain the conflicting results surrounding the relationship between inhibition skills and mathematics performance concerns the nature of mathematics involved. Rather than being a unitary skill, mathematics is a multi-componential construct. Not only can different domains be identified, for example arithmetic, algebra, or geometry, but researchers have also identified specific components that cut across these domains (see review by Rittle-Johnson and Schneider, 2014). Researchers typically discriminate factual knowledge, procedural skill and conceptual understanding. Factual knowledge comprises memorized number facts, for example the addition and multiplication tables. Procedural skill concerns the accurate and efficient execution of operations (e.g. ‘carrying’ when adding above 10) and can be thought of as ‘knowing how’. Conceptual understanding, on the other hand, is knowledge of the principles and relationships that underlie mathematics (e.g. knowing that addition is the inverse of subtraction) or ‘knowing why’. It is well established that there are complex relationships among these components (Baroody and Dowker, 2003), which are not hierarchically ordered. Furthermore, individuals differ in their profile of performance across these components, and may have strengths in one component but not others (Dowker 2005) suggesting that these components rely on differential sets of skills.
The vast majority of previous research exploring the role of inhibition has involved general standardized or curriculum tests of mathematics, which do not capture these individual elements. However, we need to move beyond these general tests of mathematics to allow the processes by which inhibition supports mathematics performance to be understood. It is likely that the precise mechanisms by which good inhibition skills support mathematics performance differ for factual, procedural and conceptual knowledge (Cragg and Gilmore, 2014).
Considering procedural skills first, there is evidence to suggest that inhibition is important in suppressing inefficient, but well rehearsed, strategies in favour of more efficient or new strategies. Lemaire and Lecacheur (2011) found that children with better inhibitory control made more use of the most efficient strategy to solve arithmetic problems compared to children with lower levels of inhibitory control. Thus, children with lower levels of inhibitory control may be able to generate alternative strategies without difficulty, but are less able to switch between strategies flexibly in response to context (c.f. Bull and Scerif 2001 performance on Wisconsin Card Sorting Task). The ability to make adaptive strategy choices is a characteristic of children who are proficient with mathematics (Torbeyns et al. 2006) and thus this mechanism might explain the general advantage in mathematics shown by children with good inhibition skills.
Turning next to conceptual understanding, Robinson and Dubé (2013) found that children with lower levels of inhibitory control made more use of mixed procedural and conceptual approaches than children with higher levels of inhibitory control, and suggested that poor inhibition was associated with an inability to suppress procedural in favour of conceptual approaches. Consistently favouring procedural over conceptual strategies could interfere with the developmental of rich conceptual understanding. Similarly, computational models of conceptual development have proposed that inhibition is required to shift attention away from procedural solutions to allow underlying numerical relationships to be identified (Siegler and Araya, 2005).
Finally, it has also been suggested that inhibition is important for correctly retrieving known number facts from memory due to the way in which number facts are stored. It has been proposed that addition and multiplication number facts are stored in an associative network (Campbell et al. 2011). As a result, the solutions to alternative problems can interfere with the retrieval of a desired solution and inhibition is required to suppress these alternatives. This can either occur from interference of “neighbouring” solutions (e.g. “42” might interfere with retrieving the answer to “6 × 8”) or alternative operations (e.g. “15” might interfere with retrieving the answer to “5 + 3”).
There is some evidence to support the hypothesis that inhibition skills are differentially related to multiple components of mathematics. Winegar (2013) found that, for children in Grades 3 and 4, inhibition skills were related to arithmetic word problem solving but not to calculation skills. Similarly, in a study of preschool children Lan et al. (2011) found that inhibition was an independent predictor of counting but not calculation skills. These differences may reflect a stronger role of inhibition in executing sequential procedures rather than recalling number facts.
These studies provide some evidence to suggest that inhibition skills play a varying role in different components of mathematics; however, these relationships have yet to be systematically explored in a single study. Furthermore, the nature of these relationships may change as children mature and develop more advanced knowledge of mathematics. The domain-general processing demands of, for example number fact knowledge, will change as children move from fragile memory of a small set of number facts through to having secure memory of a complete number fact table. It is important, therefore, to consider how these relationships change over development.
The present study
Here, we present a study that explores the relationship between inhibition skills and mathematics performance in detail by addressing three key research questions. First we explore whether inhibition skill measured in both numerical and non-numerical contexts is related to overall mathematics performance. Secondly, we consider whether inhibition skill is related to the individual components of factual, procedural and conceptual knowledge of mathematics as well as overall achievement. Finally, we test whether the relationship between inhibition skill and different components of mathematics is consistent across ages. To answer these questions, we administered numerical (dot comparison) and non-numerical (animal stroop) versions of inhibition tasks as well as tests of individual components of arithmetic knowledge and overall mathematics achievement to adult and child participants.