Introduction
 
Developmental dyscalculia (DD) is a developmental learning disorder affecting the acquisition of school-level arithmetic skills. It is a disorder with serious consequences for life success, as low numeracy skills increase the probability of unemployment, depression, physical illnesses, and even arrest (Parsons & Bynner, 2005). Likewise, it is a crucial predictor of the ability of both patients and health professionals to appropriately use health-care related information, such as dosage of medicine or health statistics (Golbeck, Ahlers-Schmidt, Paschal, & Dismuke, 2005; Hibbard, Peters, Dixon, & Tusler, 2007; Peters, Hibbard, Slovic, & Dieckmann, 2007).
 
Despite being as common as developmental dyslexia (see entry by Ruth S. Shalev on  Identification, Classification, and Prevalence of Developmental Dyscalculia), DD has not achieved the same level of recognition from the public, policymakers, or researchers. As recently as 2007, studies on dyslexia outnumbered those on dyscalculia by 14:1 (Mazzocco, 2007).
 
However, an interest in and acknowledgment of the disorder are ever increasing, and recent years have seen a productive growth in research into the causes and characteristics of DD. The present section of the Encyclopedia brings together the summaries of seven experts on the topic of DD, presents the current state of the field and highlights avenues for future progress.
 
What is Dyscalculia?
 
Experts Shalev and von Aster discuss issues related to how to recognise and classify DD. First, despite recent progress in our understanding of dyscalculia, significant debate still surrounds how to best diagnose DD, because many researchers still use different tests and different selection criteria. In turn, such inconsistency leads to wide variations in estimates of how common the disorder is. Shalev and von Aster suggest that using absolute performance measures (e.g., lowest 10 percent of children), in combination with assessing how easily an individual’s problems may be improved through teaching intervention, might be the most robust approach at present. This is in contrast to the method of looking at the difference between arithmetic and IQ, for example (so called ‘discrepancy measures’).
 
Another key issue is that a range of problems other than a dysfunction of mathematical brain networks might be responsible for poor arithmetic performance. ADHD, mathematical anxiety, poor teaching methods, large class sizes and family adversity have all been shown to impact negatively on arithmetic performance, and should be taken into account when diagnosing dyscalculia. The range of potential influences on mathematical ability has lead some researchers to suggest there may be subtypes of DD, stemming from verbal (language based processes such as the understanding of sentence structure and grammatical rules, which apply to organising numbers into a ‘mathematical language’) versus spatial processing (the ability to mentally represent and manipulate the position of items in space) deficits (Rourke, 1989, 1993). However, inconsistent results mean this research has not contributed to advances in understanding or treating DD.
 
Despite the challenges facing the basic classification of DD, research over the last ten years has reversed the opinion that DD is a rare disorder.  A range of international, large-scale studies now suggest that DD is at least as common as developmental dyslexia, affecting approximately 3-14% of the population.
 
Dyscalculia and Other Developmental Disorders
 
In her entry, Rubinsten points out that so-called ‘pure’ deficits in mathematical processing are seen in a minority of DD children. DD frequently co-occurs with other disorders, in particular dyslexia, and understanding the sources and characteristics of co-morbidity is key to developing knowledge and treatment for the disorder. The fundamental problem is, simply put, why do these disorders co-occur so frequently? Is there some common factor which underlies these disorders, or are there simply several different disorders, arising from different deficits, which sometimes overlap?  
 
Children with ADHD often exhibit problems in calculation, and this may be because generalised attention impairments make it difficult to attend to the detail required for successful arithmetic performance. On the other hand, some children with ADHD may simply have specific numerical deficits that co-occur but are not related to their ADHD. With regards to co-existing (comorbid) dyslexia, there may be a distinction in the type of math deficits shown by pure DD children and comorbid children. Pure DD children progress faster in the development of computational skills, perhaps as a result of being able to use language skills to compensate for poor numerical abilities. However, several recent studies have failed to show such group differences when using low level numerical processing measures, suggesting that the differences between pure and comorbid dyscalculia may emerge at more complex levels of cognitive processing.
 
Dyscalculia and Brain Development
 
To understand the causes of and treatment pathways for DD it is essential to understand how brain mechanisms, which support numerical processing in typically developing children, function in children with DD. By understanding the action of such brain mechanisms, we gain access to a level of insight that sometimes eludes purely behavioural data. In other words, while DD children may show similar behavioural performance to typically developing children on some measures, the brain mechanisms which support that performance may differ, giving clues to possible compensatory mechanisms. Additionally, brain imaging data aids in the process of localising behavioural deficits, which may be caused by a range of potential deficits, to more specific cognitive systems.
 
In his entry, Ansari summarises the current state of knowledge on this topic. Neuroimaging research suggests that when healthy adults process numerical quantity or ‘magnitude’, they engage bilateral regions of the parietal lobe, specifically, the intraparietal sulcus (IPS). A second parietal region in the left cerebral hemisphere, the angular gyrus, is reliably involved when adults perform formal calculations, as opposed to simply comparing or processing quantities. The relatively few developmental neuroimaging studies of numerical cognition which exist at present suggest that as children age, they increase the degree to which they engage the above parietal regions relative to frontal cortex regions during numerical processing. This is not to say that by adulthood, only the IPS and angular gyrus are involved in numerical processing and calculation, as the prefrontal cortex and other regions continue to be involved. The exact nature of the ‘fronto-parietal shift’ that Ansari describes, whether related to the representation of magnitude, or to task performance, has yet to be fully resolved.
 
Only a handful of studies have thus far investigated the function of these ‘fronto-parietal’ numerical processing networks in DD children. Ansari and Cohen-Kadosh in their contributions to this section review the existing literature and conclude that the most reliable finding across studies is that there is some degree of functional or structural disruption of parietal brain regions, particularly the IPS, in DD children during numerical processing.
 
As Cohen Kadosh points out, some of the reviewed studies find the principle disruption of IPS function in DD to be in the left hemisphere, while others find it in the right hemisphere. These findings, while apparently contradictory, may suggest the existence of subtypes of DD, as highlighted by Shalev, that are evident at the brain level. However, large numbers of further studies are required before that idea can be confirmed. Cohen Kadosh also highlights that despite the outstanding questions regarding which hemisphere of the brain the principle dysfunction is located in, the IPS does appear to be the primary brain area implicated in DD, however, the source of this deficit is still unclear. Some theorists argue that it reflects a core deficit of ‘number sense’ that is innately specified and present from birth in typically developing individuals. This ‘number sense’ is an intuitive understanding of what numbers mean, in terms of their size and how they relate to other numbers, either in terms of order or relative size. Therefore, an impaired ‘number sense’ would mean that an individual struggles with the most basic understanding of numbers, such as knowing that eight is larger/more than five. It is also possible, however, that DD stems from a failure of the parietal cortex to build a concept of number from lower level non-numerical features of the environment, such as size, density and area. In this idea, the ‘number sense’ is not in-built at birth, but rather emerges or is ‘constructed’ from our interactions with the environment as we develop, and in some children with DD, the parietal cortex fails to construct a number sense effectively. The only way that these two opposing viewpoints can be resolved, however, is through large-scale longitudinal research. That is, research which follows and regularly tests a large group of children (ideally from birth) over the course of many years. While such projects do exist in terms of behavioural research, no such project has yet been established which investigates questions about the brain basis of numerical abilities and DD in a truly longitudinal way.
 
Intervention
 
Experts Wilson and Räsänen highlight some of the challenges facing the development of effective interventions for numeracy disorders. In particular, high cost and practical difficulties mean that many intervention studies are of poor quality with limited general applicability.
           
A key barrier to effective intervention research identified by Wilson and Räsänen is that such research typically investigates diverse populations. Different classification criteria used to identify children as dyscalculic makes it impossible for the field to reach a consensus on issues of characterisation and remediation. This point, also highlighted by Shalev in her contribution to this section, is one of, if not the, principle hindering factor currently slowing progress in DD research. Furthermore, Wilson and Räsänen highlight the fact that numeracy/mathematical ability is not single skill, but a collection of component processes, and one or more of these processes may be independently impaired. Thus, they conclude, tentatively, given the limited availability of quality research on the topic that optimal interventions should take into account the composite nature of numeracy, and tailor their approach to the specific impairments of the individuals in their population.
 
Future Directions and Implications for Policy and Service Provision  
 
It is clear from each contribution to this section of the Encyclopedia that while much exciting progress is being made in understanding what causes DD, and how to remediate it, much more work is needed. More research is fundamental if we are to begin to understand why DD co-occurs so frequently with other disorders, whether it is a single disorder stemming from one specific deficit, or a collection of subtypes, and what intervention methods are most effective. Perhaps the key to this is focusing on low level cognitive processes, such as numerical magnitude processing (often measured by asking people to decide which of two numbers is numerically larger), before advancing to the direct study of higher ‘school-level’ achievement outcomes, which are typically multi-faceted and complex measures of cognitive ability. The first step, however, is for researchers to establish a working consensus on how to operationally define DD. A growing body of research reveals qualitative differences between children with pure DD and those with less severe mathematical learning difficulties (e.g., Mazzocco, Feigenson, & Halberda, 2011), and thus it is counter-productive for researchers to continue to equate the two groups through the use of liberal classification criteria. If DD is to be recognised by policymakers internationally, and appropriate funding provided for future research, then a coherent, unified body of research must be presented, and this depends first of all on adopting a consensus on classification criteria.
 
Implications for Educators
 
The contributions reviewed above give rise to several implications for educators and caregivers dealing with children with DD and mathematical learning difficulties. However, these implications are not solutions. What we can conclude is first, there are a range of factors stemming from both the individual and the environment that can negatively impact mathematical ability. Thus, when assessing children with DD, it is imperative to take into account the broader picture of the individual’s skill set, as well as their home and educational environments, and to use this information to tailor the given intervention. This is a highly complex challenge, but one that skilled educators are qualified to meet given the right training. This point, however, is not something novel to most educators, as the complexity of the classroom environment and individual children’s ability profiles is rarely reflected in basic scientific research.

Second, although a range of factors may impact math ability, there is at least a subset of children whose problems appear to stem from a brain level deficit in the ability to understand and process basic numerical information (number sense). It is important to recognise that such a specific learning disorder does exist, and that these children’s problem may run deeper than those whose problems stem from environmental factors. One way of separating these two groups, and identifying the ‘true dyscalculics’, is to go beyond curriculum based testing, which tends to focus on higher level arithmetic skills, and instead investigate very basic numerical processing skills. For example, DD children have been shown to be slower and less accurate in such seemingly simple tasks as numerical magnitude comparison (comparing which of two numbers is larger) and simple number naming (Landerl, Bevan, & Butterworth, 2004).
 
Finally, it should be noted that while work is underway to develop effective interventions for DD, that much, much more research is required before effective educational applications can be derived. Thus, one of the principle implications that can to be drawn from the above work is that patience is required, from educators and scientists alike. It will take time to translate basic research results into methods that can be applied in complex classroom environments, but with patience, and through continued discussion and interaction between researchers and educators, great progress can be made.