Is this Dyscalculia? New perspectives on diagnosing mathematical difficulties

Twelve-year-old Maria was asked to divide 38 by 2. She drew 38 crosses on the page, drew a circle around each one and then painstakingly alternated between drawing an arrow pointing to the left on one circle and an arrow pointing to the right on the next until the page became a great muddle of overlapping arrows and circles. Then she tried to count all the ones that had an arrow pointing to the left but quickly got confused by all the marks on the page and gave up in despair. She crumpled up the page and burst into tears. Despite above average performance in many areas of the curriculum, and maths teaching that had been effective for the rest of her year group, Maria had immense difficulty in acquiring basic arithmetic skills. Is this dyscalculia? 

The question of how to differentiate dyscalculia from other difficulties with mathematics has been debated for many years. Is it best viewed as a categorical disorder or just a useful label for attainment that lies at a particularly low point on a continuum of mathematical difficulties? What are its core defining features? This week, the Specific Learning Difficulties Assessment Standards Committee (SASC) published guidance to support assessors in reaching diagnostic conclusions when assessing mathematics. The aim of the guidance, based on recent research and the consensus of a working group, was ‘to provide updated, evidence based, operationally effective definitions and procedures on the assessment of specific learning difficulties in mathematics’. Central to the guidance is a definition of a Specific Learning Difficulty (SpLD) in Mathematics, with dyscalculia identified as a sub-category within it..

This definition follows swiftly on the heels of the Delphi definition of dyslexia published last month and its conceptual framework is similar in many ways. Maths difficulties meeting the diagnostic criteria for an SpLD in Mathematics are described as a set of processing difficulties which occur unexpectedly and are persistent, though change in their manifestation and impact over time. As with dyslexia, a difficulty with fluency is seen as a key feature, multiple interacting factors are viewed as affecting outcomes, and co-occurrence with other neurodevelopmental conditions is acknowledged as being more common than not. A holistic approach is encouraged, including a thorough history taking which considers a full range of factors that may put someone at risk of mathematical difficulties or give them resilience.

As in SASC’s previous guidance on assessing for dyscalculia and maths difficulties, assessors are urged to assess both ‘domain-specific’ factors which are particular to maths learning and ‘domain-general’ factors which have a more general effect on learning.

Identification of difficulties in numerical magnitude processing is essential for a diagnosis of dyscalculia, according to the new SASC guidelines. An impairment in numerical magnitude processing will present differently across the life-span but is likely to include difficulties with:

• Naming, ordering and comparing physical quantities and numbers
• Magnitude estimation
• Place value
• Counting

Numerical magnitude processing as the core domain-specific factor in dyscalculia replaces ‘number sense’ in the previous definition published by SASC, largely due to widespread inconsistencies in research and practice in the use of the term ‘number sense’, and limitations in its potential for distinguishing between dyscalculia and maths difficulties arising from other specific learning difficulties. Although numerical magnitude processing can be for symbolic information (Arabic numerals) or non-symbolic (e.g. dots), it is the capacity for symbolic information processing that seems to be the stronger predictor of mathematical performance.

Those who have specific and persistent difficulties in mathematics but do not meet the criteria for dyscalculia as they do not have extreme difficulties with numerical magnitude processing, can be diagnosed with an SpLD in Mathematics. It is noted in the definition that this can be as debilitating as dyscalculia, which again breaks away from the idea of dyscalculia as being at the lowest point on a continuum of difficulty.

An SpLD in Mathematics is likely to arise from a combination of domain-general variables and domain-specific factors. Based on the work of De Schmedt (2022), SASC lists the following possible domain-general factors that might be relevant: language, working memory, inhibitory control, task shifting, spatial skills and fluid intelligence. Domain-specific factors include specific mathematical vocabulary, mathematical anxiety, and the home and school mathematics environments.

Key markers of an SpLD in Mathematics are listed as being difficulties in arithmetic fluency and flexibility, and mathematical problem solving.

Earlier SASC guidance recognised the need to assess verbal reasoning skills to get a full cognitive profile, and the ability to solve worded maths problems to investigate limitations in language that might be having an impact. The new guidance, however, explicitly identifies language as both a domain-general and domain-specific factor affecting the mathematical learning. In addition to including verbal reasoning as an area to be tested, it recommends assessment of receptive language and listening comprehension, and expressive language. Receptive language is important for understanding instructions and the tasks themselves, as well as explanations. Expressive language is important for explaining mathematical thinking. It is also noted that it might be relevant to assess phonological awareness, especially in younger learners as there may be a link between maths and phonology in terms of the pattern-recognising skills needed, as well as its more obvious relevance to learning digit names and other terminology. The mastery of mathematical language is considered to be the ‘domain-specific’ factor relevant to progress in maths. Many linguistic terms are required, some of which are specific to maths and others which are also used more generally e.g. those related to quantity, time and space. Language is important for conceptual understanding and retention. It can be thought of as the ’container’ for a concept and is an important part of numerical cognition.

Automaticity in completion of mathematical tasks has long been considered an important indicator of proficiency and earlier SASC guidance recommends completion of mathematical tasks under both timed and untimed conditions, to investigate this. The new guidance, however, gives the concept of fluency more explicit and central emphasis, and explores it more comprehensively. In addition to automaticity (often measured in terms of a combination of speed and accuracy), flexibility is recognised as being an important component. This is the ability to select and adapt strategies appropriately to solve problems. 

The following areas are specifically mentioned for inclusion in an assessment, with examples of the types of tasks that could be used:

• Speed of processing and retrieval.
• Verbal working memory, short term phonological memory, inhibitory control and shifting attention.
• Visual-spatial processing.

Although mentioned in previous guidance, the role of executive functions in influencing mathematical performance has been given more emphasis in this new document. Working memory has long been recognised as being required for many aspects of maths, but there is now more emphasis on factors such as inhibitory control and the ability to shift attention, in creating the cognitive flexibility that is so important for maths. 

Within the guidance there is a detailed information about the knowledge, understanding and skills required to conduct assessments to investigate the presence of an SpLD in Maths or dyscalculia. There are three main components:

• Knowledge and understanding of mathematical skills.
• Teaching qualifications and experience.
• Training, knowledge and experience of holistic SpLD assessment and its specific application to Maths.

Within this new guidance, online assessment of mathematical difficulties is discouraged. ‘An online assessment should only be considered in exceptional circumstances where it is in the best interests of the individual assessed and there is not option for a face-to-face assessment’. It is stated that an assessment of need may be more appropriate than an assessment that results in a diagnostic conclusion.

Qualitative information is essential to any assessment of mathematics, including details of patterns of errors, strategies used, conceptual understanding, psychological factors (e.g. anxiety, confidence, impulsivity and motivation). For a diagnostic assessment however, reliable, valid standardised assessments must also be used. 

SASC’s SpLD Test Evaluation Committee (STEC) is preparing a detailed list of approved tests to use for different components of a SpLD in Mathematics assessment. The following tests are currently on the 2024 list for testing various areas of attainment and cognition which have been highlighted in the new guidance. 

For assessing maths attainment, the Academic Achievement Battery (AAB) and the Feifer Assessment of Mathematics (FAM) are on the current SASC list, with the FAM also being on the list for exploring sense of number. For assessing language, the current list has the Oral Passage Understanding Scale (OPUS) for receptive language. Up to age 11, the language tests within the Intelligence and Development Scales – 2^nd edition (IDS-2) can be used to assess receptive language, expressive language and phonological awareness. There are also tests of verbal reasoning up to age 20.11.

The IDS-2, already on the STEC/SASC list for assessment of cognition, though not specifically for mathematics, has tests of many of the domain-general areas of cognition specified in the new guidance as being required for the assessment of SpLD in Mathematics, and dyscalculia. These include tests of speed of processing and retrieval (Listing words, Processing speed), visual motor speed tasks (Visual motor skills), verbal working memory (Auditory short-term memory tests), other executive functions tests (Divided attention, Animal colours, Planning routes), visual-spatial processing tests (Picture recall, Visual spatial working memory, Abstract reasoning, Visual processing). The Logical Mathematical Reasoning subtest can be useful for exploring mathematical reasoning and problem-solving, as well as for assessing language and verbal reasoning in relation to mathematics through discussion with the test-taker.