Given the well-documented failings in mathematics education in many Western societies, there has been an increased interest in understanding the cognitive underpinnings of mathematical achievement. Recent research has proposed the existence of an Approximate Number System (ANS) which allows individuals to represent and manipulate non-verbal numerical information. Evidence has shown that performance on a measure of the ANS (a dot comparison task) is related to mathematics achievement, which has led researchers to suggest that the ANS plays a critical role in mathematics learning. Here we show that, rather than being driven by the nature of underlying numerical representations, this relationship may in fact be an artefact of the inhibitory control demands of some trials of the dot comparison task. This suggests that recent work basing mathematics assessments and interventions around dot comparison tasks may be inappropriate.

The process by which adults develop competence in symbolic mathematics tasks is poorly understood. Nonhuman animals, human infants, and human adults all form nonverbal representations of the approximate numerosity of arrays of dots and are capable of using these representations to perform basic mathematical operations. Several researchers have speculated that individual differences in the acuity of such nonverbal number representations provide the basis for individual differences in symbolic mathematical competence. Specifically, prior research has found that 14-year-old children's ability to rapidly compare the numerosities of two sets of colored dots is correlated with their mathematics achievements at ages 5-11. In the present study, we demonstrated that although when measured concurrently the same relationship holds in children, it does not hold in adults. We conclude that the association between nonverbal number acuity and mathematics achievement changes with age and that nonverbal number representations do not hold the key to explaining the wide variety of mathematical performance levels in adults.

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.

Since the time of Plato, philosophers and educational policy-makers have assumed that the study of mathematics improves one's general ‘thinking skills’. Today, this argument, known as the ‘Theory of Formal Discipline’ is used in policy debates to prioritize mathematics in school curricula. But there is no strong research evidence which justifies it. We tested the Theory of Formal Discipline by tracking the development of conditional reasoning behavior in students studying post-compulsory mathematics compared to post-compulsory English literature. In line with the Theory of Formal Discipline, the mathematics students did develop their conditional reasoning to a greater extent than the literature students, despite them having received no explicit tuition in conditional logic. However, this development appeared to be towards the so-called defective conditional understanding, rather than the logically normative material conditional understanding. We conclude by arguing that Plato may have been correct to claim that studying advanced mathematics is associated with the development of logical reasoning skills, but that the nature of this development may be more complex than previously thought.

Research has demonstrated that children and adults have an Approximate Number System (ANS) which allows individuals to represent and manipulate the representations of the approximate number of items within a set. It has been suggested that individual differences in the precision of the ANS are related to individual differences in mathematics achievement. One difficulty with understanding the role of the ANS, however, is a lack of consistency across studies in tasks used to measure ANS performance. Researchers have used symbolic or nonsymbolic comparison and addition tasks with varying types and sizes of stimuli. Recent studies with adult participants have shown that performance on different ANS tasks is unrelated. Across two studies we demonstrate that, in contrast to adults, children's performance across different ANS tasks, such as symbolic and nonsymbolic comparison or approximate addition, is related. These findings suggest that there are differences across development in the extent to which performance on nonsymbolic and symbolic tasks reflects ANS precision.