Mathematics for Dyslexics and Dyscalculics

A Teaching Handbook

Steve Chinn, Richard Edmund Ashcroft

• Publisher: Wiley-Blackwell
• Year: 2017
• ISBN:
  9781119159988
• Edition: 4
• Topic: Bildung
• Subtopic: Lernschwierigkeiten

A seminal handbook in the field for more than 20 years, this new and updated edition of Mathematics for Dyslexics and Dyscalculics contains the latest research and best practices for helping learners with numerical and mathematical difficulties.

• Provides a complete overview of theory and research in the fields of dyslexia and dyscalculia, along with detailed yet pragmatic methods to apply in the classroom
• Contains enhanced coverage of place value and the role of the decimal point, why fractions can challenge a developed logic for arithmetic, and the complexity of time along with new material on addressing anxiety, fear, motivation, and resilience in the classroom; and links to new resources including standardized tests and recommended reading lists
• Written by two mathematics teachers with 50 years of teaching experience between them, much of it in specialist settings for students with specific learning difficulties
• Offers effective teaching strategies for learners of all ages in a structured but accessible format

1
Dyscalculia, Dyslexia and Mathematics

Introduction

In 1981, when we moved from working in mainstream schools and began teaching in schools for dyslexic learners, our initial expectation was that teaching mathematics would be much the same as before. At that time we could not find any source of guidance to confirm or contradict this expectation. We thought dyslexia meant difficulties with language, not mathematics. Experience would, very quickly, change this impression.

Over the last 35 years, and the 23 years since we published the first edition of this book, we have accumulated experience, tried out new (and old) ideas, researched, read what little appropriate material was available (there is still far less published on learning difficulties in mathematics than on language (Gersten et al., 2007)), learned from our learners and have become convinced that difficulties in mathematics go hand in hand with the difficulties of dyslexia and, especially, that a different teaching attitude and approach is needed.

The first four chapters of this book look at some of the background that influenced the evolution of these teaching methods and continues to underpin their ongoing development. This requires a look at the learner, the subject (mathematics), the teacher and the pedagogy. The main mathematical focus of this book is number, primarily because this is the first area of mathematics studied by children and thus provides the first opportunity to fail. Our experience suggests that number remains the main source of difficulty for most of the learners we have worked with, even in secondary education. We also know that the foundations for all work to GCSE (the national examination for 16‐year‐old students in England), and beyond, are based in these early learning experiences. The evaluations and expectations of a child’s mathematical potential are often based, not always correctly, on performance in early work on number (e.g. Desoete and Stock, 2011). The remaining chapters describe some of the methods we use to teach our dyslexic learners, with the ever‐present caveat, that no one method will work for all learners.

One of the main reasons for the first four chapters is to address the complexity of learning profiles. This will explain why the methods described in the subsequent chapters are effective, but still will not meet the needs of every single child, and why teachers need the skill of responsive reactivity. There are now a number of researchers who have referred to this complexity and from a number of perspectives. Watson (2005) states:

There is no standard recipe for mathematical success. The joyous range of characteristics that make each child an individual ensure that this is true, so teachers need an understanding of the child and the subject to be able to adjust methods and improvise, from secure foundations and principles, to meet those individual needs.

Mabbott and Bisanz (2008) note that, ‘Children who experience difficulties in mathematics are a heterogeneous group’ and as Zhou and Cheng (2015) express so elegantly and succinctly, ‘mathematical competence is a constellation of abilities’. Kaufmann and a collection of international researchers (2013) writing together say that heterogeneity is a feature of developmental dyscalculia. Chapter 2 provides more detail on some of the reasons for this heterogeneity.

We also believe that a greater understanding of the ways dyslexic and dyscalculic students learn and fail mathematics will illuminate our understanding of how other children learn and fail mathematics. In other words, the reasons for failure are unlikely to be specific to dyslexic and dyscalculic learners. Poor performance in maths spreads beyond students identified as dyscalculic, for example Rashid and Brooks (2010) found low levels of attainment in a significant percentage of the population of 13–19‐year‐old students in England. The extrapolation from this is that many, if not all of the methods advocated in this book will also help many non‐dyslexic and non‐dyscalculic students to learn mathematics. We have long been advocates of the principle of learning from the ‘outliers’ (Murray et al., 2015).

Our aim has always been to teach mathematics in a mathematical way rather than seek out patronising collections of mnemonics and one‐off tricks.

Definitions of Dyslexia

The year 2016 marks the 120th anniversary of the publication of the first paper (Pringle‐Morgan, 1896, reproduced in the BDA Handbook 1996) describing a 14‐year‐old student with specific difficulties with reading, which Pringle‐Morgan labelled, based on Kussmaul’s study in 1878, as ‘congenital word blindness’. Pringle Morgan also described idiosyncratic difficulties for the young student in maths: ‘Interestingly he could multiply 749 by 867 quickly and correctly as well as working out (a + x)(a − x) = a² − x², yet failed to do 4 × ½.’

The issue of mathematics disappeared from definitions of dyslexia for a while, for example in 1968 the World Federation of Neurology defined dyslexia as: ‘A disorder manifested by a difficulty in learning to read, despite conventional instruction, adequate intelligence and socio‐cultural opportunity. It is dependent upon fundamental cognitive difficulties that are frequently of a constitutional character.’

But, by 1972 the Department of Education and Science for England and Wales included number abilities in its definition of specific reading (sic) difficulties. In the USA, the Interagency Conference’s (Kavanagh and Truss, 1988) definition of learning disabilities included ‘significant difficulties in the acquisition of mathematical abilities’ and, in the UK, Chasty (1989) defined specific learning difficulties as: ‘Organising or learning difficulties, which restrict the students competence in information processing, in fine motor skills and working memory, so causing limitations in some or all of the skills of speech, reading, spelling, writing, essay writing, numeracy and behaviour.’

In 1992 Miles and Miles, in their book Dyslexia and Mathematics, wrote: ‘The central theme of this book is that the difficulties experienced by dyslexics in mathematics are manifestations of the same limitations which also affect their reading and spelling.’

In 1995 Light and Defries (1995) highlighted the comorbidity of language and mathematical difficulties in dyslexic twins, one of the earliest mentions of the possibility of comorbid dyslexia and dyscalculia.

In the new millennium, it seems that the definitions of dyslexia are moving back to focus solely on language. This is likely to be due to the current interest in and awareness of dyscalculia and comorbidity and the trend in the UK to see ‘specific learning difficulties’ used as an umbrella term to cover dyslexia, dyscalculia, dyspraxia (developmental coordination disorder) and dysgraphia, rather than a label that was solely interchangeable with dyslexia. This is relevant for our perceptions of dyscalculia and mathematical learning difficulties. So, recently in the UK, the Rose Report’s (2009) definition of dyslexia focused on reading and spelling, with no mention of arithmetic or numeracy skills: ‘Dyslexia is a learning difficulty that primarily affects the skills involved in accurate and fluent word reading and spelling. Characteristic features of dyslexia are difficulties in phonological awareness, verbal memory and verbal processing speed.’ However, within the report, there are discussions on co‐occurring issues, which include difficulties with mental calculation.

In the USA, the International Dyslexia Association adopted a definition of dyslexia (2002), which also focused on language:

Dyslexia is a specific learning difficulty that is neurobiological in origin. It is characterised by difficulties with accurate and/or fluent word recognition and by poor spelling and decoding abilities. These difficulties typically result from a deficit in the phonological component of language that is often unexpected in relation to other cognitive abilities and the provision of effective classroom instruction. Secondary consequences may include problems in reading comprehension and reduced reading experience that can impede growth of vocabulary and background knowledge.

If dyslexia and dyscalculia are now to be defined as separate, distinct specific learning difficulties, then the concept of comorbidity (e.g. Cirino et al., 2015; Shin and Bryant, 2015) becomes very relevant. An important question for researchers is to decide whether the comorbidity is causal, independent or a different outcome resulting from the same neurological basis. The study by Moll et al. (2014) suggests that deficits in number skills are due to different underlying cognitive deficits in children with reading disorders compared to children with mathematics disorders. These deficits are, for reading disorders, a phonological deficit and, for mathematics disorders, a deficit in processing numbers.

Our classroom experience is that most of the dyslexics we have taught have had difficulties in at least some areas of mathematics. It should be noted that, in our school, the results from our specifically designed intervention, in terms of grades achieved in GCSE (the national exam for 16‐year‐old students in England) were from A* to D and with one ex‐student, who was severely dyslexic, obtaining a degree in mathematics. The theme of this book is of positive prognosis.

In her seminal book, Yeo (2002) looked at the issues surrounding dyspraxia, dyslexia and mathematics difficulties. The specific learning difficulty, dyspraxia (developmental coordination disorder) brings another set of issues to a pupil’s attempts to learn mathematics.

Finally in this section, we should be aware that dyslexia is a problem internationally (as dyscalculia certainly is). Although the English language is probably the most challenging language to learn, especially for the mastery of spelling, dyslexia occurs in many languages. For example, the Yemen Dyslexia Association (Al Hakeemi, 2015) defines dyslexia as: ‘A functional disorder of the left side of the brain. It causes difficulty in reading, writing or mathematics associated with other symptoms such as weakness in short‐term memory, ordering, movements and directions awareness.’

The Evolution of Definitions of (Developmental) Dyscalculia

At the time (2015) of writing this, the fourth edition of our book, the idea of a specific mathematics disability, now known as dyscalculia in the UK, had slipped out of common usage in our government documents, whereas at the time of the third edition it had recently slipped in. This observation draws attention to the influence of governments on the recognition of and provision for learning disabilities. A search for ‘dyscalculia’ on the gov.uk website on 10 April 2015 yielded no results, suggesting instead that we try to search for ‘calculi’.

The term dyscalculia remains not well defined, or at least without a consensus, though there have been some recent proposals as to what the definition should be (e.g. Kaufmann et al., 2013). However, it does now seem agreed that it is a specific learning difficulty that is solely related to mathematics, that is, there is no mention of a comorbid language difficulty. As one would expect, the prevalence of dyscalculia will be dependent on how it is defined.

It should be stated at this stage that, erroneously, for some people ‘dyscalculia’ suggests a dire prognosis, that of a permanent inability to do mathematics. This would be ‘acalculia’, a complete loss of the ability to work with numbers and caused by a stroke or a traumatic injury to the brain. The two terms are not interchangeable.

It remains the situation that much less research exists in comparison to dyslexia. When David Geary spoke at the 2002 IDA conference he compared our knowledge of dyslexia to being close to adulthood and our knowledge of maths learning difficulties to being in its early infancy. Gersten et al. (2007) give data on the ratio of papers on reading disability to mathematical learning disability for five subsequent decades. For 1966–1975 the ratio was 100:1 and for 1996–2005 it was 14:1. Desoete et al. (2004) note that from 1974 to 1997 only 28 articles on maths learning difficulties were cited in Psyc‐Info, whereas there were 747 articles on reading disabilities.

The work of Kosc, a pioneer in the field of dyscalculia, plus a review of the early literatur...