ORIGINAL ARTICLE Low numeracy and dyscalculia: identification and intervention Brian Butterworth • Diana Laurillard Accepted: 1 June 2010 Ó FIZ Karlsruhe 2010 Abstract One important factor in the failure to learn arithmetic in the normal way is an endogenous core deficit in the sense of number. This has been associated with low numeracy in general (e.g. Halberda et al. in Nature 455:665–668, 2008) and with dyscalculia more specifically (e.g. Landerl et al. in Cognition 93:99–125, 2004). Here, we describe straightforward ways of identifying this deficit, and offer some new ways of strengthening the sense of number using learning technologies. 1 Introduction One of the central problems in educational neuroscience is how to coordinate the disciplines of education and neuro- science. As (Varma, McCandliss, and Schwartz, 2008) note, this could be a long job. At the new Centre for Educational Neuroscience in London, an inter-institutional development involving UCL (University College London), the Institute of Education and Birkbeck College, both of the University of London, we see the way forward con- sisting in a continuing conversation among neuroscientists, psychologists and educators, seeking issues of common concern, agreed methods and useful applications. In this paper, we report on a domain-specific project to investigate the neuroscience of number and how to relate these find- ings to their implications for mathematical education. Perhaps the most important difficulty we face is how to evaluate the results of our research. In neuroscience and psychology, we can design an experiment, run it and get a result, which is methodologically straightforward. How- ever, education is far more complex, and getting a ‘‘result’’ is not straightforward, due to the interference of the many variables in the learning environment. In this study, instead of attempting a large-scale sum- mative evaluation of our interventions for dyscalculia, we are at this stage conducting an intensive formative evalu- ation of our methods, to look at the detailed effects on the behaviour of selected individuals, in the context of specific task activities and number concepts. The aim is to provide data on each learner’s difficulties, their behaviour on the learning activities, and the changes in their performance within the task, and thereby develop an account of the role the intervention plays, if any, for each learner’s conceptual development. The other important issue in educational neuroscience is to close the circle from neuroscience to education and back again to the neuroscience. We have just begun this process and can only indicate how this might continue in the future. In this paper, our first aim is to show how we get from neuroscience to an analysis of the core cognitive problem in dyscalculia. However, even if one can reliably identify the core problem—behaviourally, neurally, or genetically, and we have indicators for all of these—this does not uniquely determine the form the pedagogic intervention should take. This is a separate process. The science informs the design of the intervention, as the science of materials informs, but does not determine the design of a bridge. And as in engineering, where theory is silent, the design can B. Butterworth (&) Institute for Cognitive Neuroscience, University College London, 17 Queen Square, London WC1N 3AR, UK e-mail: [email protected]B. Butterworth Á D. Laurillard Centre for Educational Neuroscience, London, UK D. Laurillard London Knowledge Lab, Institute of Education University of London, London, UK 123 ZDM Mathematics Education DOI 10.1007/s11858-010-0267-4
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Low numeracy and dyscalculia: identiï¬cation and intervention
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ORIGINAL ARTICLE
Low numeracy and dyscalculia: identification and intervention
Brian Butterworth • Diana Laurillard
Accepted: 1 June 2010
� FIZ Karlsruhe 2010
Abstract One important factor in the failure to learn
arithmetic in the normal way is an endogenous core deficit
in the sense of number. This has been associated with
low numeracy in general (e.g. Halberda et al. in Nature
455:665–668, 2008) and with dyscalculia more specifically
(e.g. Landerl et al. in Cognition 93:99–125, 2004). Here,
we describe straightforward ways of identifying this deficit,
and offer some new ways of strengthening the sense of
number using learning technologies.
1 Introduction
One of the central problems in educational neuroscience is
how to coordinate the disciplines of education and neuro-
science. As (Varma, McCandliss, and Schwartz, 2008)
note, this could be a long job. At the new Centre for
Educational Neuroscience in London, an inter-institutional
development involving UCL (University College London),
the Institute of Education and Birkbeck College, both of
the University of London, we see the way forward con-
sisting in a continuing conversation among neuroscientists,
psychologists and educators, seeking issues of common
concern, agreed methods and useful applications. In this
paper, we report on a domain-specific project to investigate
the neuroscience of number and how to relate these find-
ings to their implications for mathematical education.
Perhaps the most important difficulty we face is how to
evaluate the results of our research. In neuroscience and
psychology, we can design an experiment, run it and get a
result, which is methodologically straightforward. How-
ever, education is far more complex, and getting a ‘‘result’’
is not straightforward, due to the interference of the many
variables in the learning environment.
In this study, instead of attempting a large-scale sum-
mative evaluation of our interventions for dyscalculia, we
are at this stage conducting an intensive formative evalu-
ation of our methods, to look at the detailed effects on the
behaviour of selected individuals, in the context of specific
task activities and number concepts. The aim is to provide
data on each learner’s difficulties, their behaviour on the
learning activities, and the changes in their performance
within the task, and thereby develop an account of the role
the intervention plays, if any, for each learner’s conceptual
development.
The other important issue in educational neuroscience is
to close the circle from neuroscience to education and back
again to the neuroscience. We have just begun this process
and can only indicate how this might continue in the future.
In this paper, our first aim is to show how we get from
neuroscience to an analysis of the core cognitive problem
in dyscalculia. However, even if one can reliably identify
the core problem—behaviourally, neurally, or genetically,
and we have indicators for all of these—this does not
uniquely determine the form the pedagogic intervention
should take. This is a separate process. The science informs
the design of the intervention, as the science of materials
informs, but does not determine the design of a bridge. And
as in engineering, where theory is silent, the design can
B. Butterworth (&)
Institute for Cognitive Neuroscience, University College
• ‘‘Wow! It is so amazing to see all this work that links
so perfectly to everything that I teach! I want to use
the programmes straight away with the boys! They
would be very useful in backing up all the concrete
work we do and also in supporting independent
work. I have often wished there were programmes
like this! Thank you for sharing them :)’’ (Teacher on
website)
• ‘‘These are exactly the kinds of programs my learners
need’’ (SEN teacher)
• ‘‘The nursery children this year are much more number
aware after playing with these games’’ (Primary
teacher)
In addition, because we test storyboards and initial
prototypes with teachers there are numerous comments on
layout, wording, and sequencing of tasks that result in
frequent changes to the design as it develops. We must
stress that we are still at the stage of formative evaluation
of these digital methods, as well as formative evaluation of
the progress of the learners.
7 Conclusions
The starting point for our research has been behaviour:
able children unable to learn arithmetic. This phenomenon
has been a puzzle for many teachers and many researchers.
A variety of different explanations, both expert and lay,
have been offered for why this happens, and doubtless
most of them can be documented in some poor learners.
For example, in an influential paper, Geary (1993)
argues that dyscalculic ‘‘children show two basic func-
tional, or phenotypic, numerical deficits’’:
1. ‘‘The use of developmentally immature arithmetical
procedures and a high frequency of procedural errors’’
(p. 346)
2. ‘‘Difficulty in the representation and retrieval of
arithmetic facts from long-term semantic memory’’
(p. 346)
Here, the idea is that there is a developmental progres-
sion from calculation strategies, such as counting on, to
established associations between problems and their solu-
tions. ‘‘Mastery of elementary arithmetic is achieved when
all basic facts can be retrieved from long-term memory
without error… [which in turn] appears to facilitate the
acquisition of more complex mathematical skills’’ (Geary
1993, p. 347). According to Geary, laying down these
associations in long-term memory depends on maintaining
the problem elements (for example, two addends, inter-
mediate results, and solution) in working memory. In
addition, the use of immature or inefficient calculation
strategies will risk decay of crucial information in working
memory. However, although there is some evidence cor-
relating measures of span with mathematical performance,
one study found no difference on a non-numerical task
testing phonological working memory (non-word repeti-
tion), suggesting that dyscalculic children do not have
reduced phonological working memory capacity in general,
although they may have a specific difficulty with working
Fig. 6 Dots2Track results for
four 10-year-old children with
dyscalculia. The horizontal axisrepresents the number of trials
to get all ten numbers correct
and the vertical axis represents
the time taken for each trial.
When performance reaches
criterion in speed and accuracy,
a new number is introduced
resulting in increased reaction
time. This creates the
characteristic saw-tooth pattern.
The mean RT for typical
10-year-olds is under 2 s
B. Butterworth, D. Laurillard
123
memory for numerical information (McLean & Hitch,
1999). Moreover, Landerl et al. (2004) found dyscalculia
even when matched against controls with comparable
spans. In this study, dyscalculics (who were in the bottom
2% of their age group on timed arithmetic) were also
matched on IQ. This suggests that general cognitive ability
alone is not a sufficient explanation. There is abundant
evidence now that it is possible to be excellent at arith-
metical calculation with low general IQ (see, e.g., Butter-
worth, 2006)
Another idea appealing to both lay and expert opinion is
that failures to learn arithmetic can be due to poor lin-
guistic skills. There are intuitive grounds for this. First,
most formal arithmetical learning comes through language;
second, people speak of a ‘‘language of mathematics’’;
third, it has been argued that arithmetical facts are stored in
a verbal form, so that impaired language could affect the
storage of useful facts; fourth, there is a statistically sig-
nificant comorbidity between arithmetical disabilities and
dyslexia. Dyslexia is usually a deficit in language abilities
that affects phonological processing which is known to
reduce working memory capacity (Nation, Adams, Bowyer-
Crane, and Snowling, 1999) which in turn may affect
lexical learning as well (Gathercole, 1995). These consid-
erations imply that dyslexics should have difficulty with
fact retrieval, if these are stored in verbal form, and with
multidigit arithmetic with high working memory load. The
problem with this line of argument, as discussed above, is
that, as we have seen, dyscalculics do not have reduced
working memory span. Moreover, both Shalev, Manor, &
Gross-Tsur (1997) and Landerl et al. (2004) found no
quantitative differences on tests of arithmetic, or on simple
number tasks such as dot enumeration and magnitude
comparison. Even in severe language disabilities, such as
specific language impairment, there appears to be no effect
on the basic capacities described here (Donlan, Bishop, &
Hitch, 1998), though it may affect learning arithmetic in
school (Cowan, Donlan, Newton, & Lloyd, 2005).
What is clear from recent research is that very basic
domain-specific core deficits can have a severe effect on
the capacity to learn arithmetic. Perhaps the best index of
this is the efficiency of enumerating small sets of objects
(typically dot arrays), although other indices have been
suggested, such as comparing dot arrays (Halberda et al.,
2008; Price et al., 2007). Our approach has been to use
pedagogical principles and best teaching practice to moti-
vate adaptive digital interventions designed to strengthen
these capacities.
Our research programme is part of a movement to bring
neuroscience, psychology and intervention closer together.
Methods for teachers of dyscalculic learners based on the
idea of a core deficit have already been developed (e.g.
Butterworth & Yeo, 2004). Indeed, a pioneering adaptive
digital method for strengthening the sense of number,
called the Number Race, has been available for 4 years in
pioneering research by Wilson, Dehaene et al. (2006) and
Wilson, Revkin, Cohen, Cohen, and Dehaene (2006). The
Number Race game is based on a rather different peda-
gogical approach from ours, and may be more suitable for
some learners.
Neither Wilson’s program, nor ours has been subjected
to formal summative evaluation. Clearly, this is a limitation
to any recommendation of how to proceed. However, for-
mative evaluation is in progress, and teachers and
researchers can contribute to the process through the
website.
In the near future, we would expect to use neuroimaging
to be part of summative evaluation of intervention for
dyscalculia, in the way that it has been used already to
assess the effects of phonological intervention on reading
development in dyslexics. There it has revealed that the
intervention tended to align neural activity with the pat-
terns found in typical readers, suggesting that the reading
process was becoming more typical rather than just more
accurate using atypical compensatory strategies (Eden
et al., 2004), a comparable approach to evaluate dyscal-
culia interventions is envisaged.
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