More Trouble with Maths
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 mathemati
cal 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 learn
ing difficulties and dyscalculia.
Steve Chinn was principal of two schools for dyslexic students before founding his own
school which received Beacon School status from the government and the Independent
Schools ‘Award for Excellence’. He designed and delivered the UK’s first Post Graduate
course on dyslexia and maths difficulties.
Steve’s research papers, articles and books include the award-winning The Trouble
with Maths. He compiled and edited The International Handbook of Dyscalculia
and Mathematical Learning Difficulties (2015). The fourth edition of his seminal book
Mathematics for Dyslexics was published in 2017. He has recently written catch-up mate
rials for Numicon.
Steve has lectured in 30 countries, including consultation work for the Singapore MoE.
He is Visiting Professor at the University of Derby.
www.pdfgrip.com
www.pdfgrip.com
More Trouble with Maths
A Complete Manual to Identifying and
Diagnosing Mathematical Difficulties
Third edition
Steve Chinn
www.pdfgrip.com
Third edition published 2020
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
and by Routledge
52 Vanderbilt Avenue, New York, NY 10017
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2020 Steve Chinn
The right of Steve Chinn to be identified as author of this work has been asserted
by him in accordance with sections 77 and 78 of the Copyright, Designs and
Patents Act 1988.
All rights reserved. The purchase of this copyright material confers the right on
the purchasing institution to photocopy pages which bear the photocopy icon
and copyright line at the bottom of the page. No other part of this publication may
be reproduced, stored in a retrieval system, or transmitted in any form or by any
means, electronic, mechanical, photocopying, recording or otherwise, without
prior permission in writing from the publisher.
Trademark notice: Product or corporate names may be trademarks or registered
trademarks, and are used only for identification and explanation without intent to
infringe.
First edition published by Routledge 2012
Second edition published by Routledge 2017
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
Names: Chinn, Stephen J., author.
Title: More trouble with maths: a complete manual to identifying and diagnosing
mathematical difficulties / Steve Chinn.
Description: 3rd edition. | Abingdon, Oxon; New York, NY: Routledge, 2020. |
Series: NASEN spotlight | Includes bibliographical references and index.
Identifiers: LCCN 2019054156 (print) | LCCN 2019054157 (ebook) |
ISBN 9780367862152 (hardback) | ISBN 9780367862169 (paperback) |
ISBN 9781003017721 (ebook)
Subjects: LCSH: Mathematics–Study and teaching. | Math anxiety. |
Mathematical ability–Testing. | Acalculia. | Dyslexia.
Classification: LCC QA11.2 .C475 2020 (print) | LCC QA11.2 (ebook) |
DDC 510.71–dc23
LC record available at />
LC ebook record available at />
ISBN: 978-0-367-86215-2 (hbk)
ISBN: 978-0-367-86216-9 (pbk)
ISBN: 978-1-003-01772-1 (ebk)
Typeset in Helvetica
by Deanta Global Publishing Services, Chennai, India
www.pdfgrip.com
Contents
Foreword by Professor Maggie Snowling
1 Introduction: Dyscalculia and mathematical learning difficulties: The test
protocol
vii
1
2 Diagnosis, assessment and teaching: The benefits of linking
14
3 The Dyscalculia Checklist: Thirty-one characteristics that can contribute
to maths failure
23
4 Starting the assessment/diagnosis: Getting to know the person through
informal activities
38
5 Short-term memory and working memory: Two key underlying skills that
influence learning
53
6 Tests of basic facts - addition, subtraction, multiplication and division:
Their role in mathematical learning difficulties and dyscalculia
60
7 Mathematics anxiety: Which topics and activities create anxiety
74
8 The 15-Minute norm-referenced Mathematics Test: Basic computations
and algebra designed to compare performances
84
9 Errors and the 15-Minute Mathematics Test: Recognising and
understanding common error patterns
108
10 Cognitive (thinking) style: How learners think about and solve
mathematics problems
127
11 Estimation: A key life skill used to develop more confidence
with mathematics
148
12 Mathematics vocabulary, symbols and word problems: Exploring how
they contribute to mathematics learning difficulties
152
13 Criterion-referenced (formative) tests: Focusing on identified problems
and showing how to build ongoing diagnosis into teaching
162
www.pdfgrip.com
vi
Contents
14 Speed of working: The implications of ‘doing mathematics’ quickly
171
15 Two sample reports
174
Appendix 1
Appendix 2
Appendix 3
A sample ‘teacher observations’ pro-forma
A pre-assessment pro-forma for parents/carers
Schools, colleges, institutions and individuals who provided
data for the norm-referenced tests in Chapters 6 and 8
References
Index
www.pdfgrip.com
203
205
206
208
213
Foreword
This is an astonishing book! ‘Troubles with maths’ are both varied and widespread;
difficulties with maths in school cause anxiety and not only affect educational achieve
ment but also limit career prospects. It follows that the early identification of problems
with numbers, followed by appropriate intervention, should be a priority for all indus
trialised societies. But knowledge of how to assess mathematical difficulties is poor
despite a growing evidence base on the nature and causes of ‘dyscalculia’. In this
book, Steve Chinn shares many years of expertise with extreme clarity; assessment
needs to be the first step to intervention, and the book provides a comprehensive
explanation for the practitioner of what is needed in order to properly understand
mathematical learning difficulties. At the core of the book is a test protocol that goes
well beyond a screening instrument. The assessment incorporates background infor
mation, teacher observations and a questionnaire probing maths anxiety as well as
more conventional tasks tapping basic number knowledge, number facts, numerical
operations and reasoning skills. It also includes standardised assessment procedures
that have been developed by the author, together with sample diagnostic reports,
generously shared. This book should be on the shelves of all professionals in the
field of maths education and educational assessment, and I am confident that it will
spearhead a much-needed increase in proficiency in the assessment of numeracy
skills in the field.
Maggie Snowling,
St. John’s College,
University of Oxford
www.pdfgrip.com
www.pdfgrip.com
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
OO A norm-referenced (UK sample*) 15-minute Mathematics Test for ages 7 to 59
years old
OO Norm-referenced (UK sample*) tests for the four sets of basic facts (+ − × ÷) for
ages 7 to 15 years old
OO 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)
OO A test of thinking cognitive (thinking) style in mathematics
OO A Dyscalculia Checklist
OO Informal tests for vocabulary, symbols, place value, estimation
OO A structured, exemplar test of word problems
OO Informal tests of short-term memory and working memory
OO Guidance on how to appraise the ability to estimate
OO Guidance on how to use errors and error patterns in diagnosis and intervention
OO Guidance on how to construct criterion referenced tests and how to integrate them
into day-to-day teaching
OO Case studies
OO
*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 diag
nose 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.
www.pdfgrip.com
2
Introduction - dyscalculia
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 defi
nition 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 indi
vidual’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 out
lined 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 disabili
ties. Their observations about the problems of building a diagnostic procedure around
www.pdfgrip.com
Introduction - dyscalculia
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 dis
rupted, how they interact, change over time and are affected by education.
Thambirajah (2011) has suggested four criteria for diagnosis of dyscalculia. They
are:
1. Difficulties with understanding quantities or carrying out basic arithmetic operations
that are not consistent with the person’s chronological age, educational opportu
nities or intellectual abilities.
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. There is significant interference with academic achievements and the activities of
daily living that require mathematical skills.
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 mathemat
ics 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 accu
rately 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 knowl
edge 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 address
ing 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.
www.pdfgrip.com
3
4
Introduction - dyscalculia
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 dyscal
culia. However, the research identified 112 pupils with dyscalculia. This 4.6% preva
lence 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 sub
jects. It is more than an ability to do basic arithmetic. It involves developing con
fidence and competence with numbers and measures. It requires understanding
of the number system, a repertoire of mathematical techniques, and an inclina
tion and ability to solve quantitative or spatial problems in a range of contexts.
Numeracy also demands understanding of the ways in which data are gathered by
counting and measuring, and presented in graphs, diagrams, charts and tables.
(DfEE Framework for Teaching Mathematics: Year 7. 1999)
However, as if to illustrate how we interchange words, the DfEE explained that the
National Numeracy Strategy would be implemented by schools providing a structured
daily mathematics lesson.
Wikipedia defines numeracy, more broadly, as:
the ability to reason with numbers and other mathematical concepts. A numeri
cally literate person can manage and respond to the mathematical demands of
life. Aspects of numeracy include number sense, operation sense, computation,
measurement, geometry, probability and statistics.
A different perspective was given, somewhat controversially, by Michael Girling (2001)
who defined basic numeracy as:
The ability to use a four function electronic calculator sensibly.
Finally, to focus in on a basic concept, number sense, which is a key component of
numeracy: Often we assume we know what we mean when we say ‘number sense’,
as with so many of the things we meet that are in everyday use. In fact, Berch (2005)
found 30 alleged components of number sense in the literature. I have selected the
www.pdfgrip.com
Introduction - dyscalculia
ones that are relevant to the test protocol used in this book. As a cluster they provide
a sensible, but somewhat lengthy description of number sense:
1. Elementary abilities or intuitions about numbers and arithmetic.
2. Ability to approximate or estimate.
3. Ability to make numerical magnitude comparisons.
4. Ability to decompose numbers naturally.
5. Ability to use the relationships among arithmetic operations to understand the
base-10 number system.
6. A desire to make sense of numerical situations by looking for links between new
information and previously acquired knowledge.
7. Possessing fluency and flexibility with numbers.
8. Can recognise benchmark numbers and number patterns.
9. Can recognise gross numerical errors.
10. Can understand and use equivalent forms and representations of numbers as well
as equivalent expressions.
11. Can represent the same number in multiple ways depending on the context and
purpose of the representation.
12. A non-algorithmic feel for numbers.
13. A conceptual structure that relies on many links among mathematical relation
ships, mathematical principles and mathematical procedures.
14. A mental number line on which analogue representations of numerical quantities
can be manipulated.
15. A nonverbal, evolutionarily ancient, innate capacity to process approximate
numerosities.
The assessment and diagnostic tools in this book primarily address arithmetic, the
part of mathematics that focuses on numbers and the four operations, that is, addi
tion, subtraction, multiplication and division. I am working on the hypothesis (and long
experience) that this is where the majority of mathematical learning difficulties are
rooted, certainly at the dyscalculia level.
Mathematical learning difficulties
If dyscalculia is at the severe end of a spectrum of mathematical learning difficulties,
then there are going to be difficulties above that imprecisely defined threshold, hence
the use of the term ‘Mathematical Learning Difficulties’. These difficulties, like dyscal
culia, stretch beyond school into adult life which suggests that they are perseverant
and/or resistant to current teaching methods.
The term ‘Mathematics Learning Disabilities’ is often used inter-changeably with
‘developmental dyscalculia’ in the USA (for example, Mabbott and Bisanz, 2008).
In the USA Mathematics Learning Disabilities are estimated to affect 5% to 8% of
school-aged children (Geary, 2004). There is, again, a difference between the use of
the word ‘disability’ and the use of the word ‘difficulty’. In this book I use the term
‘Mathematical Learning Difficulties’ and am using that to refer to the bottom 20–25%
in terms of achievement in mathematics.
www.pdfgrip.com
5
6
Introduction - dyscalculia
The Programme for International Student Assessment from the Organisation for
Economic Co-operation and Development (PISA) collects and publishes data for
maths. New results on their international assessments are due as I write this (2019),
but they will not be available before my publisher’s deadline. The most recent results
are from 2015. A sample of the positions in the survey has Shanghai (1), Singapore (2),
Hong Kong (4), Switzerland (9), New Zealand (16), UK (24), Ireland (28) and the US (35).
The UK’s results for 15-year-olds in maths have remained stable since 2006, around
the OECD average.
Rashid and Brooks (2010) in their study, ‘The levels of attainment in literacy and
numeracy of 13- to 19-year-olds in England, 1948–2009’, noted that 22% of 16- to
19-year-olds are functionally innumerate and that this has remained at the same level
for at least 20 years.
The 2017 National Numeracy booklet, ‘A New Approach to Making the UK
Numerate’ stated that ‘Government statistics suggest that 17 million adults – 49% of
the working-age population of England – have the numeracy level that we expect of
primary school children’.
The UK was the worst performing of the 17 OECD countries in the ‘Numeracy/
Knowledge’ component of Adult Financial Literacy (2016). Mathematics as taught in
UK schools seems not to endure into adulthood.
Further evidence of the persistence of the problem with low achievers in math
ematics comes from Hodgen et al. (2010) and their 30-year comparison of attainment
in mathematics in secondary school children:
A further rather worrying feature is that in all three topic areas (algebra, deci
mals and ratio) and all year groups there are now a higher proportion of very
low performances than there were in 1976/7. It is difficult to explain this; one
possibility is the closing of many Special Schools and greater inclusivity within
the mainstream sector. However it is not clear whether this factor could account
for the full size of the difference. Another possible explanation lies in the finding
that the National Numeracy Scheme introduced into schools in 1999 had the
effect of depressing attainment at the lower end, perhaps because of the failure
to address children’s particular needs in attempting to provide equal access to
the curriculum.
Many of the problems surrounding mathematics are international, for example,
Ramaa and Gowramma (2002) found that 25% of the children in their sample of
1408 primary-aged Indian pupils were considered by their teachers to have arithmetic
difficulty
Learning difficulties in mathematics can be caused by many factors, with each
factor contributing a variable influence that depends on many things, such as the
mathematics topic or the current level of anxiety in the individual. Some of the fac
tors can be attributed to the person, for example, a poor working memory; some are
external, for example, inappropriate instruction. (More detail is provided in Chapter 2
of the companion book, The Trouble with Mathematics: A Practical Guide to Helping
Learners with Numerical Difficulties.)
Thus, learning difficulties in mathematics are a complex problem and any diagno
sis will largely reflect the situation on the day and time when it is carried out, although
www.pdfgrip.com
Introduction - dyscalculia
a thorough procedure will usually produce much useful and valid information. One
consequence of this complexity and the many factors involved is that the approach
to assessment/diagnosis should always be multi-dimensional and flexible. A second
consequence is that there will be, inevitably, a spectrum of difficulties for every factor.
This should not be a revelation to any educator. We should expect a wide variation in
children and adults and for the normal distribution to apply to each of the contribut
ing factors. It will be a heterogeneous population. Kaufman et al. (2013) argue that
heterogeneity is a feature of developmental dyscalculia.
A further implication of the heterogeneous nature of mathematics learning difficul
ties is that there should be no prescribed order of structure for the assessment or the
subsequent intervention. For example, it may be that for one person it is the anxiety
issues that have to be addressed before any input for the cognitive issues. For another
person, it may be that the investigation has to be targeted initially at a particular area
of mathematics, such as basic facts, so that that particular barrier can be overcome
and success experienced. Ultimately all the factors will interlink. However, it is often
the case that the very basics of maths need to revisited and secured before progress
can be achieved.
As a final observation for this section, I have included some (UK) data from the
norm-referencing process for the 15-minute test included in this book. The percent
ages are for correct answers.
)
10 6030
10yrs
15yrs
44.5%
62.3%
13yrs
16–19yrs
48.7%
62.6%
23 ÷ 1000
10yrs
15yrs
14.5%
46.8%
13yrs
16–19yrs
31.4%
51.2%
54.5%
81.8%
13yrs
16–19yrs
72.8%
88.3%
75.9%
83.2%
13yrs
16–19yrs
74.9%
87.5%
63.6%
83.6%
13yrs
16–19yrs
69.6%
87.9%
33
16
10yrs
15yrs
37
42
73
+68
10yrs
15yrs
103
96
10yrs
15yrs
www.pdfgrip.com
7
8
Introduction - dyscalculia
)
2 38
10yrs
15yrs
59.1%
75.0%
13yrs
16–19yrs
60.7%
74.7%
10yrs
15yrs
14.5%
46.8%
13yrs
16–19yrs
31.4%
38.4%
1
=
4 12
10yrs
15yrs
65.5%
84.0%
13yrs
16–19yrs
75.4%
88.3%
)
9 927
Write 0.125 as a fraction in its simplest form
10yrs
7.7%
13yrs
15yrs
31.4%
16–19yrs
17.3%
33.6%
20% of 140
10yrs
15yrs
32.3%
66.8%
13yrs
16–19yrs
53.9%
73.3%
150% of £64
10yrs
16.4%
15yrs
60.0%
13yrs
16–19yrs
46.1%
63.0%
541
´203
10yrs
15yrs
13yrs
16–19yrs
15.2%
39.5%
14.1%
38.2%
2y + 5 = 31
y=
13yrs
52.4%
15yrs
60.9%
16–19yrs
63.7%
(My favourite error for this question, if not the whole test was: y = years)
(x + 15) + (x − 23) = 44
13yrs
9.9%
15yrs
23.6%
x=
16–19yrs
24.6%
1 km ÷ 5 =
13yrs
15yrs
metres
37.7%
56.0%
16–19yrs
58.7%
5.67 km =
13yrs
15yrs
metres
36.6%
51.0%
16–19yrs
49.1%
www.pdfgrip.com
Introduction - dyscalculia
Tests and testing
This book contains a range of tests and diagnostic activities. They represent the most
significant causal factors for mathematical learning difficulties and dyscalculia. Each
one is included, of course, to make meaningful contributions to the assessment and
diagnosis. However, tests have to elicit answers to be of any use. The classic reaction
of the anxious, low-confidence learner is not to attempt the task (Chinn, 1995). The
assessment must be carried out in a way that creates, at the very least, a basic level
of confidence and a willingness in the subject to participate.
Should you have concerns about test anxiety, there is a test anxiety inventory for
children and adolescents, the TAICA (Whitaker Sena et al., 2007).
Tests are a snapshot of ‘now’ and are influenced by many of the factors that we
may suspect are present, such as anxiety, and by factors that we may not fully appre
ciate are present, such as the memories of past experiences of learning mathematics.
We need to remember that testing is something that you do to the student, whereas
diagnosis is something you do with the student. So, this book contains a range of
materials that will help the process of diagnosis.
If you are about to undertake an assessment/diagnosis, then there are some ques
tions that might guide you in carring out that task.
Some basic questions
For each component of the assessment/diagnosis, the leading questions are:
What do you want to know?
Why do you want to know it?
How will you investigate it?
Then more targeted questions should include:
OO
OO
OO
OO
How severe is the problem?
You may want to know a mathematics age or the percentile at which the subject
is performing. This will require an appropriate norm-referenced test. Knowing the
severity of the deficit may attract resources to help address the problem.
What can’t they do?
It is important to know where the gaps are so that the intervention can be directed
efficiently. For example, there may be issues with procedures, working memory,
language or speed of working. Problems may lie a long way back in the roots of
arithmetic.
What can they do?
Intervention will be most effective if it starts where the learner is secure. The proto
col must focus on the strengths as well as the weaknesses. For example, addition
seems to be the default procedure for many people who are weak at mathematics.
Finding topics that ‘they can do’ can be used to give a rare experience of success
for the learner.
What don’t they know?
There are two facets to this question. One is from the USA’s National Research
Council report (Bransford et al., 2000) on How People Learn. It had three key findings.
www.pdfgrip.com
9
10
Introduction - dyscalculia
Part of the second finding is that students need ‘a deep foundation of factual knowl
edge’. For some ‘traditionalists’ this means, for example, that students should be able
to retrieve from memory (quickly) all of the times table facts. If ‘not knowing’ these
facts is made into an issue, then not knowing them may become part of the problem
of learning mathematics. There is also substantial evidence, for example, Nunes et
al. (2009) and Ofsted (2006), that the dominant way of teaching mathematics in the
UK is by memorising formulas. The Education Endowment Foundation (Henderson
et al., 2017) provides eight recommendations to improve outcomes in maths for
7–14-year-old students. Over-reliance on memorising is not a top recommendation.
The other facet is to define what exactly constitutes a ‘deep’ foundation of
knowledge. In other words, what do you NEED to know (and what can you work
out)?
OO What do they know?
Intervention should start where the learner’s knowledge and, hopefully, understand
ing are secure. Sometimes it may be necessary to question what seems to be
known rather than understood. A good memory can take you a long way in basic
mathematics, but understanding is better and memory alone does not seem to be
enough as mathematics progresses. There seems also to be a decline in the amount
of information stored and available for retrieval as time (undefined) moves on.
OO How do they learn?
The process of learning is more complicated than the process of memorising. I sus
pect this statement applies to teaching as well. There are many factors around this
four-word question. These may include the cognitive style of the learner (Chapter
10), their working memory (Chapter 5) and their response to teaching materials (that
is, do they need to start at the concrete stage of learning?).
OO How can I teach them?
It would be sensible to teach them the way that they learn (Chinn, 2020). Much
of this concept is covered in The Trouble with Mathematics: A Practical Guide to
Helping Learners with Numerical Difficulties (4th edition, 2020). One of the objec
tives of a diagnosis is to find out the way the student learns.
OO What does the learner bring?
A learner can bring emotional baggage, a lot of anxiety, poor self-efficacy and a
long history of failure at mathematics. In my informal survey of teachers around the
UK over many years and in many other countries, teachers are stating that enough
children are giving up on mathematics at seven years old to be noticed in a class.
So, an adult of 19 years may have many years of failure and withdrawal behind them.
OO Where do I start the intervention?
This is another key question. The answer is usually further back than you might
initially think. The answer to this question has to be another key objective for the
diagnosis to answer.
A diagnostic protocol
In 1991, Chinn suggested a structure for diagnosis that included:
1. A standardised (norm-referenced) mathematics age
2. An assessment of the child’s ability to recognise and use mathematics symbols
www.pdfgrip.com
Introduction - dyscalculia
3. An assessment of basic fact knowledge, compensatory strategies and under
standing of numbers and their relationships
4. Cognitive style
5. The level of understanding of place value
6. Mathematics language
7. A measure of the learner’s accuracy in calculations
8. An assessment of understanding and accuracy in using algorithms (procedures
and formulas)
9. A measure of speed of working
10. An analysis of error patterns
11. A test of ability to solve basic word problems
Later this was modified (Chinn and Ashcroft, 1993) to include attitude and anxiety,
money and mathematics language. Later working memory and short-term memory
were added to the list.
The structure of this protocol forms the basis for the work in this book. Of course,
it has been further refined and modified over the following 30 years.
It seems to be important that the test and procedures included in this book are
practical, user-friendly and informative. They have been chosen because they can
generate information that will help in the assessment and provide diagnostic clues
as to how intervention can best be provided for each individual. The tests and activi
ties, with the exception of the tests for working and short-term memories, are directly
about mathematics. There is no need to extrapolate any findings.
Three norm-referenced tests have been produced by the author specifically for
inclusion in this book. Also included is a test of cognitive style, which was written
by the author with colleagues John Bath and Dwight Knox and, in its original form,
published in the USA in 1986.
One of the goals of this book is to make the interpretation of tests more realistic,
maybe even intellectually cynical. The process of assessment/diagnosis is there to
help a child or an adult, not to provide data for performance comparisons and politi
cal points. It is about finding out why there are difficulties and what can be done to
address those difficulties.
Mathematical learning difficulties and individuals
The realisation that children with learning difficulties in mathematics are a heterogene
ous group is not new. For example, in 1947, Tilton noted that ‘some children fail owing
not to carelessness or simple ignorance, but because of individual misconceptions of
rules, and a lack of grasp of number concepts’. Austin (1982) observed, ‘perhaps the
learning disability population simply includes too diverse a student population to make
teaching recommendations unique to this group’. Chinn and Ashcroft (2016) consider
that the interactions between the factors that can contribute to learning difficulties in
mathematics create an enormous individuality amongst dyslexic learners.
There are a number of individual reasons and even more combinations of reasons
why a child or adult may fail in mathematics. The implication from this should be that
there will need to be both a range of interventions available to teach to those individual
profiles and an equally diverse set of diagnostic tools which are used in a responsive
www.pdfgrip.com
11
12
Introduction - dyscalculia
and adaptable diagnostic protocol. This is a heterogeneous population (Bugden and
Ansari, 2015), facing a constellation of mathematical challenges (Zhou and Cheng,
2015).
Teaching and diagnosing
The two activities, teaching and diagnosing, should be inextricably linked. Each should
inform the other, hence this volume and its complementary volume, The Trouble with
Mathematics: A Practical Guide to Helping Learners with Numerical Difficulties. One
book is on diagnosis and the other is on teaching. The relationship between diagnosis
and teaching is an example of the chicken-and-egg dilemma. The answer may not be
quite appropriate to a biologically sound solution for the chicken or the egg, but for
intervention and diagnosis, the two should be concurrent.
Co-occurring difficulties: Comorbidity
There has been an increasing awareness of co-occurring difficulties. From a per
sonal perspective, the first edition of the Chinn and Ashcroft book, Mathematics for
Dyslexics (1993) became Mathematics for Dyslexics including Dyscalculics in the third
edition (2007).
This has recently raised a question, ‘Does a diagnosis of maths learning difficulties
and/or dyscalculia necessitate a diagnosis of dyslexia as well?’
I think the answer to this question is, ‘It depends’. In other words, I do not consider
it to be obligatory. A perceptive diagnosis (which one hopes applies to all diagnoses)
would recognise behaviours that might indicate a need for a complementary diagno
sis for the co-occurrence of dyslexia. A further factor would be the reasons why the
diagnosis was requested.
A paper from a group of researchers from the US (Willcutt et al., 2013), which
includes three highly respected names in the field, looked at three groups, Reading
Disability only, Mathematics Disability only and RD + MD. Not surprisingly, the RD +
MD group exhibited the most pronounced academic and social impairments. All three
groups exhibited significant weaknesses on measures of processing speed, working
memory and verbal comprehension. Deficits in phonological processing and naming
speed were uniquely associated with reading difficulties, whereas difficulty shifting
cognitive set (an executive function which involves conscious change in attention) was
specifically associated with deficits in maths.
Landerl and Moll (2009) carried out a study which tested the hypothesis that dys
lexia and dyscalculia are associated with two largely independent cognitive deficits,
namely a phonological deficit in the case of dyslexia and a deficit in the number mod
ule in the case of dyscalculia. Again, three groups were studied, RD, MD and RD+
MD. Their findings were:
A phonological deficit was found for both dyslexic groups, irrespective of addi
tional arithmetic deficits, but not for the dyscalculia-only group. In contrast, defi
cits in processing of symbolic and non-symbolic magnitudes were observed in
both groups of dyscalculic children, irrespective of additional reading difficulties,
www.pdfgrip.com
Introduction - dyscalculia
but not in the dyslexia-only group. Cognitive deficits in the comorbid dyslexia/
dyscalculia group were additive; that is, they resulted from the combination of
two learning disorders. These findings suggest that dyslexia and dyscalculia have
separable cognitive profiles, namely a phonological deficit in the case of dyslexia
and a deficient number module in the case of dyscalculia.
Further reading
Chinn, S. (2020). Mathematics learning difficulties and dyscalculia. In: L. Peer and G. Reid
(eds.), Special Educational Needs: A Guide for Inclusive Practice, 3rd edn. London: Sage.
The chapter gives a comprehensive overview of dyscalculia.
www.pdfgrip.com
13
2
Diagnosis, assessment and teaching
The benefits of linking
Diagnosis, assessment and teaching should always be interlinked and should be
ongoing. Assessment can be used to identify the extent of the problems with learn
ing, and diagnosis should lead to ideas for understanding and addressing those
problems, but more importantly, both can be used to pre-empt many future problems
that could arise when learning mathematics. As far as is possible we need to get
teaching right first time in order to capture the powerful influence of the first learning
experience.
Assessment is an integral part of teaching and learning.
Primary School Curriculum, Ireland (1999)
And so is diagnosis.
Assessment is about measuring the student’s achievements, skills and deficits. It’s
about knowing what the student can do and her level of performance and knowledge,
usually in comparison to a peer group.
Diagnosis is about going beyond finding what a learner can and cannot do. It is
about identifying how they learn, what they can do and cannot do and what they do
when they don’t understand and understand. It’s about what they are not learning
and why they are underachieving. As a consequence, it should also lead to advice on
appropriate intervention.
Sometimes advice from the past remains pertinent today:
It is our view that diagnosis is a data collection procedure for determining instruc
tional needs.
(Underhill et al., 1980, p. 12)
Brueckner and Bond (1955) explained three levels of diagnosis:
General diagnosis is the use of comprehensive survey tests and other general
evaluation procedures. This could be seen as ‘assessment’.
OO Analytical diagnosis is the use of systematic procedures for locating and identify
ing specific weaknesses.
OO Case study diagnosis is the application of clinical procedures that provide for a
detailed study of the performance or achievement of an individual pupil with an
evident learning problem so as to determine, as specifically as possible, the nature
and seriousness of the learning difficulty and the underlying causes.
OO
Dowker (2005) suggests that looking solely from a perspective of procedural errors
can be a potential problem in diagnosis. Diagnosing an incorrect strategy should not
be the final step. Looking for conceptual reasons behind the strategy is more revealing
www.pdfgrip.com
Diagnosis, assessment and teaching
for planning interventions. Hence the use of questions such as, ‘Can you tell me how
you did that?’
The tests and diagnostic activities included in this book can be selected and
combined to access all three levels. My goal is to provide materials for a thorough
diagnosis and that includes some assessment tools.
We need to remember two influences that may help us decide which data to col
lect and how to collect it:
Whereas cognitive ability reflects what an individual can do, personality traits
reflect what an individual will do.
(Hattie, 2009, p. 45)
Assessment
Assessment may be carried out to compare the learner’s achievements with his/her
peers or to identify the mathematics they can do and understand and the mathemat
ics they cannot do and understand. The first objective can be achieved via a normreferenced test (NRT), and the second may need the additional information provided
by criterion-referenced tests (CRTs).
My first degree was in chemistry where I learnt about the Heisenberg Principle. I
am taking what it stated, on a sub-atomic scale, and applying it to testing. The prin
ciple says that what you measure you change, because the process of measuring has
an influence on the quantities you are trying to measure. On a human scale, there is
a risk that the act of measuring changes the level of response, usually negatively, and
this is often due to the process of measuring creating a debilitating anxiety.
For example, one of the bigger challenges in compiling material for this book
was the task of collecting data for the 15-minute summative test from adults. Many
adults do not want to spend up to 15 minutes doing a Mathematics Test as a favour
to someone they don’t know. I achieved a little more success with adults by renaming
the ‘Mathematics Test’ a ‘Mathematics Survey’.
Norm-referenced tests
A standardised test is administered and scored according to specific and uniform
procedures. When standardised tests are used to compare performance to stu
dents across the country they are called ‘standardised norm-referenced tests.’
(Kubiszyn and Borich, 2007)
Thus, a norm-referenced test is a test which has been given to a suitably large, rep
resentative sample of children and/or adults. The scores from the sample are exam
ined and analysed. A teacher using the test can then compare his students’ results
with the norm-referenced results. This should give a measure as to how far ahead
or behind a student’s performance is when compared to his peers. NRTs are often
used to measure progress (but see later) or the current state of a student’s learning
and achievement.
Kubiszyn and Borich, (2007) advise that ‘the information obtained from an NRT is
usually not as useful for classroom decision making as the information obtained from
www.pdfgrip.com
15
16
Diagnosis, assessment and teaching
CRTs’. If the opportunity is there, using both types of test will lead to the most useful
information.
Sometimes people use the term ‘summative evaluation’ to describe the measure
ment of achievement. Summative evaluations are usually normative.
The questionnaire below may help when selecting a norm-referenced test. There
are no definitive answers to the questions. Using the questionnaire is about helping
you to identify the test that most closely matches your specific needs.
Questionnaire for selecting a norm-referenced test (NRT)
Starting with five general questions:
1.
2.
3.
4.
5.
What do you need from the test?
Is it for screening a group?
Is it part (hopefully not all) of a diagnosis?
Is it to measure progress?
How old is the student/subject?
And then 13 more specific questions:
OO Is
it a test which is restricted in use to psychologists (that is, not available for
teacher use)? For example, the WISC, the Weschler Intelligence Scale for Children,
may only be used by psychologists. There are some sensible reasons for this to
be the case.
OO How much does it cost (a) initially and (b) for extra test sheets and score sheets?
OO What does it look like? For example, are the items too close together? Is the
font clear? Does its appearance overwhelm the subject? Is it age-appropriate in
appearance? (See below.)
OO How much time does the test take? There will be a compromise between a test
being thorough enough to give a valid measurement and being too daunting, thus
taking up too much of the often limited endurance span of the subject and con
sequently not leaving enough time to investigate other facets and components of
learning because the subject is now too tired mentally to perform at a level that is
typical of his/her ability.
OO What age range is the test designed for?
In terms of curriculum content, a norm-referenced test is by its nature a sampling
test. It should include enough items from across the curriculum to give a picture
of overall achievement. Setting up a test could be viewed as a statistical exercise
in that the items that are included in the test are there because they contribute
to the normal distribution of scores for the test, not just because they are test
ing a particular procedure (which would make them criterion-referenced items
as well). By restricting the age range, the range of topics can be reduced which
can make the test shorter or make it more thorough in examining the topics it
does cover.
There is a potential problem in using restricted age ranges. If the student is,
say, 15 years old, but working at the level of ten-year-old, the test that matches
the achievement level may look as though it has been designed to interest a
www.pdfgrip.com
