More Trouble with Maths

A Complete Manual to Identifying and Diagnosing Mathematical Difficulties

Steve Chinn

• Publisher: Routledge
• Year: 2020
• ISBN:
  9781000052060
• Edition: 3
• Topic: Pedagogía
• Subtopic: Educación general

Now in an updated third edition, this invaluable resource takes a practical and accessible approach to identifying and diagnosing many of the factors that contribute to mathematical learning difficulties and dyscalculia. Using a combination of formative and summative approaches, it provides a range of norm-referenced, standardised tests and diagnostic activities, each designed to reveal common error patterns and misconceptions in order to form a basis for intervention. Revised to reflect developments in the understanding of learning difficulties in mathematics, the book gives a diagnostic overview of a range of challenges to mathematical learning, including difficulties in grasping and retaining facts, problems with mathematics vocabulary and maths anxiety.
Key features of this book include:

• Photocopiable tests and activities designed to be presented in a low-stress way
• Guidance on the interpretation of data, allowing diagnosis and assessment to become integrated into everyday teaching
• Sample reports, showing the diagnostic tests in practice

Drawing on tried and tested methods, as well as the author's extensive experience and expertise, this book is written in an engaging and user-friendly style. It is a vital resource for anyone who wants to accurately identify the depth and nature of mathematical learning difficulties and dyscalculia.

1 Introduction

Dyscalculia and mathematical learning difficulties: The test protocol

This book was written to complement The Trouble with Mathematics: A Practical Guide to Helping Learners with Numeracy Difficulties. It looks at assessing and diagnosing and learning difficulties in mathematics and dyscalculia and links those processes to the teaching philosophies and pragmatics in The Trouble with Mathematics: A Practical Guide to Helping Learners with Numerical Difficulties (now in 4th edition).

It contains:

• A suggested diagnostic protocol and the reasons for selecting the components
• A norm-referenced (UK sample*) 15-minute Mathematics Test for ages 7 to 59 years old
• Norm-referenced (UK sample*) tests for the four sets of basic facts (+ − × ÷) for ages 7 to 15 years old
• A norm-referenced (English sample*) anxiety, ‘How I feel about mathematics’, test of mathematics anxiety for ages 11 to 16 years old (a version for adults is available on my website, www.stevechinn.co.uk)
• A test of thinking cognitive (thinking) style in mathematics
• A Dyscalculia Checklist
• Informal tests for vocabulary, symbols, place value, estimation
• A structured, exemplar test of word problems
• Informal tests of short-term memory and working memory
• Guidance on how to appraise the ability to estimate
• Guidance on how to use errors and error patterns in diagnosis and intervention
• Guidance on how to construct criterion referenced tests and how to integrate them into day-to-day teaching
• Case studies

*Samples for each test were over 2000

Many of the factors from the protocol interact.

The tests and procedures in this book should enable teachers and tutors to diagnose and identify the key factors that contribute to learning difficulties in mathematics and dyscalculia. There are many examples where the relationships between topics reinforce the need to take a broad and flexible approach to diagnosis and assessment.

None of the tests are restricted.

Dyscalculia

This book is about assessing and diagnosing mathematics learning difficulties and dyscalculia. It takes the view that mathematics learning difficulties are on a spectrum. At the severe end of the spectrum, the learning difficulties might be labelled as ‘dyscalculia’.

This book is also about the evidence that might be collected, evaluated and analysed to make decisions about those mathematics learning difficulties, their causes and their severity.

A definition of dyscalculia, a specific learning difficulty, published by the UK’s Department of Education (2001) stated:

Dyscalculia is a condition that affects the ability to acquire mathematical skills. Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers and have problems learning number facts and procedures. Even if they produce a correct answer, or use a correct method, they may do so mechanically and without confidence.

Note that this is not a deficit definition.

The DSM-5 definition from the American Psychiatric Association offered this definition of (developmental) dyscalculia (2013):

A specific learning disorder that is characterised by impairments in learning basic arithmetic facts, processing numerical magnitude and performing accurate and fluent calculations.
These difficulties must be quantifiably below what is expected for an individual’s chronological age and must not be caused by poor educational or daily activities or by intellectual impairments.

It is of note that this is a deficit definition and that it also rules out poor educational activities as a root cause. This definition tallies well with the diagnostic protocol outlined in this book, where many of the tests are to probe for unexpected low scores. Further support for the existence of difficulties could be gleaned from the responses to adequate and standard teaching.

Note that ‘acquired dyscalculia’ is the consequence of brain injury or stroke.

In the UK, SASC (2019) published this definition of dyscalculia:

Dyscalculia is a specific and persistent difficulty in understanding numbers which can lead to a diverse range of difficulties with mathematics. It will be unexpected in relation to age, level of education and experience and occurs across all ages and abilities.
Mathematics difficulties are best thought of as a continuum, not a distinct category, and they have many causal factors. Dyscalculia falls at one end of the spectrum and will be distinguishable from other mathematics issues due to the severity of difficulties with number sense, including subitising, symbolic and non-symbolic magnitude comparison, and ordering. It can occur singly but often co-occurs with other specific learning difficulties, mathematics anxiety and medical conditions.

Kavale and Forness (2000) wrote a critical analysis of definitions of learning disabilities. Their observations about the problems of building a diagnostic procedure around a definition, as I think is proper, lead me to think, as a lapsed physicist, that a fully satisfactory definition of dyscalculia has yet to evolve. Towards this future, Bugden and Ansari (2015) discuss the emerging role of developmental cognitive neuroscience in helping us to discover much more about the precise parts of the brain that are disrupted, how they interact, change over time and are affected by education.

Thambirajah (2011) has suggested four criteria for diagnosis of dyscalculia. They are:

1. 1. Difficulties with understanding quantities or carrying out basic arithmetic operations that are not consistent with the person’s chronological age, educational opportunities or intellectual abilities.
2. 2. The severity of the difficulties is substantial as assessed by standardised measures of these skills (at or below the fifth percentile of achievement) or by academic performance (two school years behind peers) and is persistent.
3. 3. There is significant interference with academic achievements and the activities of daily living that require mathematical skills.
4. 4. The arithmetic difficulties are present from an early age and are not due to visual, hearing or neurological causes or lack of schooling.

There are a couple of observations to make with these criteria:

• The use of the word ‘chronological’ does not imply that mathematics achievement levels continue to increase throughout our age span, but it is more relevant to the age of students when at school.
• The choice of the fifth percentile is somewhat arbitrary, but does match the general/average prevalence quoted in research papers on dyscalculia, for example, Ramaa and Gowramma’s (2002) study found that 5.54% of their sample of 1408 children were considered to exhibit dyscalculia.
• Prior to this Kavale (2005) had discussed the role of responsiveness to intervention in making decisions about the presence or absence of specific learning disabilities. It is evident that many children do not respond to more of the same, even when delivered ‘slower and louder’.
• In the US, Powell et al. (2011) defined low performance in mathematics as mathematics difficulty, where low performance is below the 26th percentile on a standardised test of mathematics. This definition is apposite for this book.
• Mazzocco (2011) defines MLD, mathematics learning disability, ‘as a domain-specific deficit in understanding or processing numerical information, which is often and accurately used synonymously with developmental dyscalculia’. Maybe it’s the replacement of ‘difficulty’ from Powell et al. with ‘disability’ that distinguishes between these two definitions and clarifies the difference in prevalence. The use of the words ‘difficulty’ and ‘disability’ is, obviously, highly significant. And there is a potential for confusion in using ‘MLD’ unless it is clear which of these two words is represented by the ‘D’.
• Bugden and Ansari (2015) add a note of caution about the current state of our knowledge about dyscalculia: ‘It is evident that current findings in the DD literature are contradictory and that there is no clear conclusion as to what causes DD. Furthermore, there is no universally agreed upon criteria for diagnosing children with DD’. However, this should not prevent schools from observing and addressing the deficits in the key skills that will depress the mathematical achievements of learners. Problems should not always require a label before they are addressed.

Recent research from Northern Ireland, Morsanyi et al. (2018), on the prevalence of dyscalculia found that of the 2421 primary school pupils in the study, 108 had received an official diagnosis of dyslexia, but only one pupil had been diagnosed with dyscalculia. However, the research identified 112 pupils with dyscalculia. This 4.6% prevalence is in line with previous research. They also noted that of this 112, 80% had other developmental disorders, such as dyslexia. There is a section on this (comorbidity) at the end of this chapter.

What is mathematics? What is numeracy?

It is valuable to know what we are assessing, whether it is arithmetic, mathematics or numeracy. These are terms that we often use casually. Although that ‘casually’ is adequate in most cases, it may be useful to look at some definitions of these words. The task is not as easy as I had hoped. Authors of ‘mathematics’ books often avoid the challenge. For a subject that often deals in precision, the definitions are not a good example.

In England we frequently use the term ‘numeracy’. We introduced a ‘National Numeracy Strategy’ for all schools in the late 1990s, defining numeracy as:

A proficiency which is developed mainly in mathematics, but also in other subjects. It is more than an ability to do basic arithmetic. It involves developing confidence and competence with numbers and measures. It requires understanding of the number syst...