Mathematical cognition: a window onto deep questions

Many adults lack the mathematical skills they need to succeed and flourish in the modern world. In the UK, for example, the OECD Skills for Life Survey has estimated that 24 per cent of adults have numeracy levels below that needed to function in everyday life, for example, to understand food pricing or pay household bills (OECD Skills for Life Survey, 2013). These difficulties arise early, with a quarter of children failing to reach the expected level of mathematics skills by the end of primary school. These limitations are associated with reduced employment opportunities, income and quality of life.

Mathematical cognition researchers believe that a better understanding of underlying mathematical processes and the influences on these can help us to identify why so many individuals struggle with mathematics, and provide insight in how to improve mathematical performance. It's against this backdrop that we wrote An Introduction to Mathematical Cognition with the hope that providing an accessible introduction to key theories, notable findings and ongoing debates would support the inclusion of this topic in undergraduate and postgraduate psychology courses and therefore introduce a new generation of researchers to the field.

Mathematical cognition is an interdisciplinary field, drawing on psychology, education, and neuroscience.

But having a positive impact on mathematics education and performance is just one important motivation for studying mathematical cognition. Understanding the processes of mathematical performance is an ideal test case to explore deeper psychological questions. Mathematical performance involves integrating formal, uniquely human, systems of symbolic representation (i.e. number words and digits) with more basic, intuitive (and some suggest innate) systems for representing mathematical ideas.

The study of mathematical cognition therefore provides a window onto deeper questions of how our cognitive systems draw on and are influenced by multiple sources of information. These questions go beyond the domain of mathematics and help us to understand the nature of our cognitive systems themselves.

Growth of the field

Mathematical cognition researchers are interested in how mathematical understanding and performance develops from infancy to adulthood, the factors which explain individual differences in mathematical performance and why some individuals find mathematics so difficult. This requires an insight into a broad range of topics including underlying brain processes, cognitive systems and their development, and social and environmental influences on learning. As a result, mathematical cognition is an interdisciplinary field, drawing on psychology, education, and neuroscience.

Methods include neuroimaging, cognitive and behavioural experiments, intervention studies, computational modelling, animal research, cross-cultural studies, as well as qualitative research. Indeed, a strength of the field is the use of a variety of methodologies to develop and test theories of mathematical processing and learning. Topics within mathematical cognition therefore provide an opportunity to demonstrate to psychology students how evidence from different methodologies can be combined to advance theories.

Despite the broad significance of mathematical cognition across different domains of psychology, the topic has only become common in the general psychology literature in the last 15 years or so. Indeed, until relatively recently it was common for the wider relevance of work on mathematical cognition to be queried by editors of generalist psychology journals. For instance, during a pre-2010 review process for one of our articles on the symbol grounding problem – the issue of how numerical symbols acquire their meaning – an editor remarked 'I found your manuscript well written and clear but I was not entirely convinced of the broad appeal of the underlying issues.

I would ask you spend a bit more time setting out why your question is of sufficient interest to be published in a general journal'. The symbol grounding problem is a fundamental question in mathematical cognition and it would be inconceivable to imagine an editor writing something similar today, showing how far mathematical cognition has come in the last decade.

The changing attitude to mathematical cognition as a valuable topic in psychology is reflected in its inclusion in textbooks. For example, Mitchell and Zielger's Fundamentals of Developmental Psychology, added a chapter on numerical development to the second edition (2013).  Despite this, it is a topic that appears on few undergraduate psychology courses in any depth.

It is not listed as a specific topic within any of the subject areas included in the BPS standards for accreditation and QAA Subject Benchmark Statement for Psychology, unlike, for example, language and language development. However, given the breadth of methodologies and topics that mathematical cognition research encompasses it is relevant to all the subject areas included in these standards (biological, cognitive, developmental, social psychology and individual differences) and an example of how these subjects can be integrated. It also provides an example of how basic psychological research can have relevance for and impact on everyday life.

Problems commonly seen in children with mathematical difficulties in primary school are counting errors, inaccurate calculations, slow or less number fact retrieval.

The chapter in our book on 'Individual differences and mathematical difficulties' discusses two topics that might be of particular interest to psychologists: mathematical difficulties and mathematics anxiety. Both are common and persist into adulthood.

Mathematical learning difficulties

Research into the causes of mathematical difficulties has significantly increased over the last two decades. However, despite being estimated to be as common as reading difficulties, it is still comparatively under-researched. Estimates of its prevalence, symptoms and causes vary significantly with the way mathematical difficulties are defined. Problems commonly seen in children with mathematical difficulties in primary school are counting errors, inaccurate calculations, slow or less number fact retrieval, and use of strategies for calculation that are typically seen in younger children (e.g. counting on fingers).

In the Diagnostic Statistical Manual of the American Psychiatric Association mathematical difficulties are listed under 'Specific learning disorders'. The DSM V definition is 'A neurodevelopmental disorder of biological origin manifested in learning difficulty and problems in acquiring academic skills markedly below age level and manifested in the early school years, lasting for at least 6 months; not attributed to intellectual disabilities, developmental disorders, or neurological or motor disorders.' Notably, the same definition is used for reading difficulties.

Individual differences in academic skills such as reading, writing and mathematics are strongly correlated across the curriculum, so in the early days the specificity of mathematical difficulties had been questioned. But today there is clear evidence – e.g. from Robin Peterson and colleagues – that despite some overlap with other academic skills, children can experience specific and isolated difficulties in mathematics.

In recent years, in line with modern conceptions of reading difficulties, a dimensional approach for mathematical difficulties has emerged. In this conceptualisation mathematical abilities are normally distributed in the population and children with mathematical difficulties are located at the lower end of this distribution. While this conceptualisation has strong empirical support, it causes some practical questions, for example, how to choose the cut-off between low mathematical skills and mathematical difficulties, including the severe mathematical difficulties often called developmental dyscalculia.

Currently there are two broad classes of causal theories of mathematical difficulties. Domain-specific theories suggest that an early deficit in foundational numerical skills causes mathematical difficulties. Several domain-specific deficits have been put forward over the last decade, including children's ability to make approximate non-symbolic comparisons, children's activation of magnitude when processing numerical symbols such as digits, children's ability to easily switch between different number formats (i.e. between spoken number words (e.g. 'three') and digits (e.g. '3')) and children's ability to judge whether numbers are correctly ordered by their size.

Domain-general theories, in contrast, suggest that mathematical difficulties are related to deficits in skills outside the numerical and mathematical domain but that are essential for mathematical competence, such as for example visuo-spatial working memory, verbal working memory, inhibitory functions or attentional functions.

Heterogeneity is one key characteristic of mathematical difficulties. This could at least partly be due to the high comorbidity between mathematical difficulties and other developmental difficulties. Statistically speaking mathematical difficulties are more likely to occur in children together with other developmental difficulties such as reading difficulties, attention-deficit / hyperactivity disorder, developmental language disorder or developmental co-ordination disorder.

It is an important question for future research whether the domain-general deficits found in children with mathematical difficulties might rather be related to their co-occurring other developmental difficulties than be the cause of their mathematical difficulties.

Parental mathematics anxiety only negatively affected children's mathematical performance when parents did not belief mathematics to be an important subject.

There are still many unanswered important questions for the field of mathematical difficulties such as how and when children with mathematical difficulties can be best identified, what are the best ways to support children with mathematical difficulties at different ages and how the common comorbid difficulties affect identification and intervention for children with mathematical difficulties. Furthermore, there are many adults struggling with mathematical difficulties, often without a diagnosis or even awareness and with very little support.

Mathematics anxiety

Mathematical competence is also influenced by affective components such as individual attitudes and feelings towards mathematics. One example we discuss in our book is mathematics anxiety. This is often defined as 'the feeling of apprehension and discomfort when using numbers or solving math problems'. With sensitive measures, mathematics anxiety can already be detected in children in early primary school. The extent and prevalence of mathematics anxiety increases over the school years. Mathematics anxiety is negatively associated with mathematical achievement.

The direction of this relationship has been debated for many years: the deficit model proposes that low mathematical performance leads to higher mathematics anxiety, while the debilitating anxiety model suggests that having mathematics anxiety interferes with the ability to successfully perform in mathematics and results in poor mathematical achievement. It looks likely that both processes can be at play.

Mathematics anxiety has negative on-line and off-line effects. In the on-line effect the person experiencing high mathematics anxiety needs extra cognitive resources to manage the negative emotions and worries arising during mathematical activities, and thus has lower cognitive resources to engage with the mathematical task at hand. The off-line effect is that individuals with high mathematics anxiety tend to avoid mathematics, mathematics-related situations and career paths that require mathematics. Both effects have far-reaching consequences.

Some risk factors for developing mathematics anxiety – such as low mathematical ability, low working memory capacity, children's perception of their own mathematical ability or susceptibility to public embarrassment – are unique to the individual child. Other factors seem to be related to parental beliefs and the educational environment.

Recent research, for example, showed that parental mathematics anxiety only negatively affected children's mathematical performance when parents did not belief mathematics to be an important subject. Teachers can also play a critical role. For example, teacher's own confidence in their ability to teach mathematics seems to be acting as a buffer with higher confidence leading to a reduction of mathematics anxiety in their students.

Open questions and future of the field

As befits a rapidly developing research area, mathematical cognition has many unanswered questions and areas that are ripe for further research. Back in 2014 we organised a small conference with the aim of producing a research agenda focused on the biggest challenges facing mathematical cognition researchers. Following an established method successfully used in other fields, a group of 16 international researchers produced a list of what they considered to be the 26 most important unanswered questions in mathematical cognition. These were published in 2016 in an article in the Journal of Numerical Cognition.

Looking back on these questions reveals that progress has been made on some – including the issues of dyscalculia and mathematics anxiety – whereas others remain almost totally open. For instance, the field has not made much progress with understanding the cognitive processes used by mathematical experts during their work, or on what mathematical intuitions of correctness/doubt are and where they come from.

Moreover, the mathematical cognition as a field is still heavily weighted towards understanding numerical processing rather than mathematical processing more generally. The relationship between this important work on foundational numerical skills and other non-numerical areas of mathematics, or with more advanced mathematical skills, remains largely unexplored.

Because of this we were keen to include chapters on logical reasoning and mathematical proof in An Introduction to Mathematical Cognition. We hope that mathematical cognition continues to develop, and that our book will have made a small contribution towards this.