Learning Difficulties in Numeracy – Characteristics of Dyscalculia (3/5)


Welcome to the third of the five
videos that have been developed in collaboration
between the University of Melbourne and the
Victorian Department for Education and Training
to address numeracy learning difficulties
and specifically dyscalculia. This video focuses
on dyscalculia as a specific maths learning deficit
as discussed in video 1. There are many different
forms of MLD including delayed number competence. Dyscalculia is considered
to be a neurological deficit affecting number sense
or number knowledge that results in an inability
to connect number symbols and words
with non symbolic forms of number, and with the spatial
aspects of number everywhere in our environment, and this is important
in two ways. They’re everywhere in that
they can cause someone with dyscalculia to be put off
by them but they’re also everywhere so that we can use
them to assist dyscalculic students
and children to access numbers more often within their
environment and I’ll raise that later. So for example you might be
asked this sort of question: how much more is the basketball
and soccer ball? It’s in newspapers, in the advertisement section
in reductions on objects that are for sale in shops,
on cafe menus, road signs, and other sorts of fast
food menus. So numbers are everywhere. For us to access
and to manipulate. Now what is dyscalculia? I’ve talked in earlier videos
that it’s hard to define. So I’m going to use
this Department for Education and Skills definition
from the UK because it characterises
dyscalculia very much in the same way
as we would regard it as a condition that affects
the ability to acquire arithmetical skills, dyscalculic learners may
have difficulty understanding simple
number concepts, like an intuitive grasp
of numbers and have problems learning number
facts and procedures even if they produce a correct
answer or use the correct method. They may do so mechanically
and without confidence. The impact of dyscalculia
is covered in an “all in the mind” programme,
which I’ll talk about later, but the link is there if you’re
interested in listening to this program
which characterises dyscalculia and describes its
forms and the impact it has on people’s lives
and in the first video I introduced this concept:
does maths matter? I talked about the way
in which it impacts young people, that is schoolchildren
and school students, but now we’re going to look at
some of the other impacts for adult life, and so students with dyscalculia
are at risk of early school leaving, low participation in post
school education and training, poor employment outcomes,
and also social isolation, and this is part of a study
by Parsons and Vita and they showed as well
from their data that numeracy problems impact
more negatively on job prospects than do
literacy problems, and in the order of four
times as many of their participants were in the very
low numeracy category compared to the very
low literacy category. Twenty seven percent versus
seven percent for women and 19 percent vs.
5 percent for men. So what are the consequences
of dyscalculia for adults? Well they can be blocked
from certain professions. There are some professions
for which number knowledge is going to be
critical. Accountancy, for example, but more importantly it can
have an impact on everyday life. The number of times that we need
to compare statistics, so for instance, if we want to make some
decisions about which interest rate we’re going
to we’re going to lock in for our mortgages or how
we’re going to organise our superannuation. All of those number competencies
come into our decision making as adults. Difficulty managing
money is a major issue. There is some evidence
that this difficulty managing money and feeling
embarrassed about it can cause some people to prefer
to shoplift rather than be embarrassed at the till
about not knowing how much to pay or how much
change they’re going to get, and in video one we’ve talked
about low self-esteem anxiety and avoidance
in school students, but it also extends
into adults as well. So these are some of the things
that adults say, “I’ve always had difficulty
with simple addition and subtraction
since young, always still have to count
on my fingers quickly, for example, for five
plus seven, without anyone knowing.” “Sometimes I feel very
embarrassed especially under pressure.
I just panic, I struggled through school
with maths to the point teachers gave up on me. I can only count on my fingers
or with a calculator. I can’t count out change
no matter how small and often get flustered with any
task involving numbers. Despite trying hard, I could never remember my
times tables. Division just bewildered
me totally. English was one of my
best subjects at school. I have no trouble whatsoever
reading or writing, understanding literary
concepts and theories but spend an hour sitting in the
bank trying to work out how much money
is in my checking account.” “Last year I returned
to university attempting to avoid any subjects
containing mathematics but hidden in nearly everything
are formulas and calculations”. So we’re now going to look
at three short videos that that feature
Brian Butterworth in a conversation with a student
with dyscalculia, with the teacher
of that student, and also with a businessman
with a lifetime lived experience of dyscalculia.
“I definitely don’t like maths, definitely not maths.” “How about we start
with what you do like?” “I do like science and history.” “I would like you to tell me
about your favourite things, what are those?” “Dinosaurs.”
“Yes, I especially love dinousaurs.” “There’s a type
of pachysepharasaur called draco rex and its name
means Dragon King and amazingly all dinousars can
link their family tree back to a very
small insect eating lizard! There were no mammals in the age
of the dinosaurs…” “I would like you to fill
in the number that is missing. So what have we got here? Twenty seven plus something
equals thirty. Yes, now twenty seven plus?” “Three.”
“Right now the next one.” He he’s just written down
two as being what you need to add to eight to make
thirty and now his teacher is asking him to build
up eight to thirty using rods and unit cubes,
white unit cubes, and you can see he’s having
trouble doing that. “What are you going to add?” He really doesn’t have much
confidence or much skill in putting out the right number. So it really takes quite
a lot of encouragement and hits from the teacher
for him to get this right. “How many are you short of?” So I’d just like to raise
a couple of points from that video. The first relates to the opening
comments by the student. The one thing I really
dislike is maths, that is actually why
I am where I am today. I was a school principal
and I was interviewing students coming into year seven
and I would always ask them what they particularly
liked about school and they would answer art
or history or what goes on in the yard at recess
time and then I would say is there anything that you don’t
like about school, and almost invariably
the answer maths would come up and I found
this quite concerning as a long term maths teacher
that so many of the students coming into the
school disliked maths. The other point that I’d
like to make about this is how obviously
intelligent this boy is. Often we associate most
learning difficulties with a perception
that the student isn’t intelligent, but you could see he had
a perfectly good memory, had a memory for all sorts
of details about dinosaurs. He obviously also liked history
and science. So his maths number difficulty
was very specific and very obvious when he
was doing quite a simple question and having
such difficulty. His teacher now talks. “The really important thing
is to establish what the concept
is and this can be terribly slow and you’ve got to…
the way we teach in England, you have confusion between
cardinal and ordinal numbers. They’re taught both from the
word go and say if children can
count to say five, they don’t understand
that the quantity of five includes one two three and four. They simply think
it’s the fifth one and you see you need to work
with concrete materials and then move from the concrete
to a pictorial and written only once they’re able to do
it with concrete materials and really understand
what they’re doing, and it’s vitally important
that the student talks because they need
to reason through it. I think you’re saw
with the children we were filming but when
they start to reason through they can
unscramble some of this.” So two important points
from those comments: the use of concrete materials
for as long as possible while students instantiate their
conceptual understanding of number is emphasized but also
the need to listen to students as they problem solve. That is exactly the strategy
we’re going to raise in the intervention video 5,
this need for the student to work through the problem
and articulate the way that they are working through
the problem so that we understand what it is
they don’t understand. Now this last video is a
business man with a lived experience with dyscalculia. “So I was basically
useless at maths. I just couldn’t do it and my
teachers couldn’t understand, I was very good at all
the other subjects but couldn’t do maths
and they thought I was deliberately stupid
or lazy or something similar. There have been so many problems
which I can’t admit, I couldn’t admit, as I was fairly successful but I
couldn’t count. So I kept it hidden until
I was 55 and then I came out, as it were, because of you Brian
and it took me such a long time to understand
and deal with this issue. I know it’s real. I have had dyscalculia
my whole life. It’s not some American fad, I know because there are so many
aspects of my life even, counting with my fingers
under a table at a board meeting. When you’re raising
millions of pounds, you know I’ve done that. I know what a real
impediment it is, and that’s why I think
it’s so important to recognize what it is and
what you can do to cope with it, and that’s why
it’s so important. I think your work is that you’re
saying look these children are not stupid. They have a specific problem
which can be identified and there are coping mechanisms.
There’s no cure, but there are coping mechanisms
and I think I’m a living embodiment of that. There’s many things I’ve done
but I’m still, by the way, I’m still
a lousy businessman. I do lots of things and I
achieve a lot that, I don’t make much money. So what
is it about maths then? Maths is componential. We’ve talked about
this in the past. It is different parts of maths
add together to give you a comprehensive
understanding of number, and this starts in the
non-verbal area so in number in terms
of approximation and comparison, skills that newborns we have
shown already have and then we move from that
into a verbal form of number, we have number facts we move
into the operations of addition, multiplication, then we move
into a logical realm where we’re looking at problem
solving and higher maths and they might look like this
algebra equation to solve or a world
problem to solve, and another aspect of number
is the spatial aspect that’s been mentioned
in earlier videos, when talk about number line
estimation and it can include geometry
and number line. Now is it possible to explain
developmental dyscalculia in relation to a cognitive model
of normal or expected function? So is it just part
of that continuum or is it something completely
different? Also, is there any evidence
of interdependence and the establishment
of these components or is it possible that some can
be acquired in a different order or in a different
capacity than others and then are there any
limitations to functional plasticity in relation
to dyscalculia? Now what do I mean
by functional plasticity? The brain is known
to have this enormous capacity for plasticity where
new pathways can be built, new connections made,
and new networks created, so that new learning can occur.
Now the question is: is that possible to dyscalculia? If dyscalculia is a neurological
deficit in the brain, is it possible for us
to create new networks so that children and students
with dyscalculia can make the same sort of progress
as other students and then is there a single
route to competence or are there alternative pathways? So there are some big questions
we have to ask with dyscalculia and some of the experimental
studies with maths learning difficulties that were run
by Geary showed two basic functional or phenotific
numerical differences and the first one
was it was the use of developmentally immature
arithmetic procedures and a high frequency of procedural errors, and if you think back
to video 2 where we talked about the development
of counting and we showed how immature strategies can
be used to achieve the end but it doesn’t necessarily… its not the most efficient
and it can lead to errors. So for many MLD children
this appears to be associated with developmental delay
in acquiring conceptual knowledge underlying
the procedural applications even though other potential
factors like working memory can’t be ruled out of this
but this is a point that will raise several
times this issue of procedural application
leading to conceptual knowledge. The second difference
that Geary noticed was that this was a much more
fundamental that is to say it doesn’t disappear
with development in number representation and also
it also included the ability to retrieve
arithmetic facts from long term memory. Now the impact of this
is a disruption of the ability to spatially
represent numerical information as in the number line that we’ve
talked about earlier and also visual spatial neurological
deficit that affects both functional skills and conceptual
understanding of number representation. Now this is really important
because one of the issues we find with dyscalculia
students is we can teach them something and they seem to learn
it and then they forget it, and this is usually because they
have acquired some sort of a process or procedure
but they haven’t conceptualized what underpins that procedure.
These are important deficits. So the two differences though
distinguish between maths delay as we talked about in the first
slide of video 1 and maths deficit and they underline
this difference between maths learning difficulties more
generally and dyscalculia specifically. So from a cognitive
perspective some lower order deficits and MLD
students potentially reside in five component skills. There are procedural
and memory retrieval, so these are applying
particular procedures to problem solve and remembering
number facts and operations along the way, and then we have these other
components which are critical for making progress beyond
the most basic number processing. Conceptual understanding is one
of those, if you don’t understand
what the procedure is about or why it’s there
then you are going to forget what the procedure is, and that requires your working
memory your working memory is when you are solving
problems and you have to hold numbers in your
head and manipulate them. That is a feature of working
memory and of course speed of processing is going
to also affect this because if you are slow
in your speed of processing then that process of working
memory is going to be slow and you’re going to forget
things that you needed to record in order to move forward. So we might divide these
into two groups then, functional skills
are the top ones, they’re evident in the process
of solving arithmetic problems so just procedural applications
and remembering things, and then we have these other
skills that might underlie and also contribute
to the procedural memory components. If you understand
what the process is, you have a conceptual
understanding of it, then you are much more
likely to remember it next time you meet a problem
like that. So what are the big questions
then that we might ask with dyscalculia? Well one question we might
ask is what causes dyscalculia and I’m hoping I’m going to be
able to answer some of these questions through the course
of this video and future videos, and what can
Educational Neuroscience tell us about it? How can it be
identified early? Can it be prevented? What is the best type
of remediation and are these subtypes and if so do
they need different remediation approaches? Now the last two of those
questions are going to be answered I hope in the
intervention video five and identified early where I’m
hoping we’re going to be able to answer that in the video
four when we’re looking at assessment and we have already
mentioned in video one preschoolers differences
in understanding number but we do need to know how
maths works in the brain. Now no video presentation
would be complete without a picture of the brain
but I can assure you this is the only one
and it is quite simple to understand. Now I talked earlier about
this connection between the non symbolic form of a number, the word and the symbol
and how important that is. Now that connection actually
happens in the brain. So it is the coming
together of different information in the brain
from the visual cortex from the memory, and what happens is that
connection is made in an area of the brain
known as the intra parietal sockets and the entire
parietal socket is just here, just above your left ear
and it is a part of the brain that is responsible
for integrating information in the brain and how we know
this is because it is more active when students
are undergoing magnitude comparison tasks where
these sorts of bits of information are essential
in order to solve the task, and we also know that there
is less grey matter, there is less brain activity
in that interparietal sockets for dyscalculics. So it does look like this
is the site of the neurological deficit
in the brain and it is this particular area
which in dyscalculic children isn’t functioning and we need
to address that by the way we intervene to try and avoid
needing that particular connection by building
new connections and of course I talked earlier
about this issue of plasticity, the brain is very plastic.
Can we form new networks? Can we make new connections
so that these students can progress with
that no understanding? Now I need to just talk about
dyslexia compared to dyscalculia. I talked earlier about
the similar prevalence rates although we are now
looking as though, now that we’re more accurate
in diagnosing dyscalculia, looks like it may be up
as high as 12 percent, but there are two very
different neurocognitive systems for literacy
and numeracy. Different brain systems
I’ve just shown you the brain system
for numeracy it is quite different for literacy. Literacy is parasitic
on language that is to say it’s not an innate system
for reading and spelling. So what language is learned
and reading and spelling is taught afterwards, but that isn’t a part
of the brain, and we talked earlier
about the evolutionary advantage of number ability, there isn’t a similar
evolutionary advantage for being able to read and write. There are biological basic
numerical capacities and these seem to be innate
as compared to this dyslexia, and we’ve already shown
that infants have quite well defined numerical
capacities right at birth. It is conceivable as well given
that there is a neurological deficit in dyscalculia
that there are different genetic bases for dyslexia
dyscalculia although we do find it occurring together, but nevertheless it also
does occur separately. So it does look like there
are different genetic bases, and then some of the questions
we get asked: is dyscalculia just due
to poor language abilities? If the students can’t read
the questions that they’re being asked
is that what the problem is? Well no, because when we compare
dyscalculia it’s with controls. We find that we have the same
color naming response time. Is it a consequence
of slow reading? No, reading disabled students
with dyslexia are normal on maths test both on in terms
of accuracy and the time they take. Is dyscalculia a consequence
of a reading disability like dyslexia? No, because the pattern
of dyscalculia is the same in those that are pure
dyscalculia and those have in addition dyslexia.
Is it a matter of low IQ? Well you saw the video with the
young lad from England working through maths. No, when you compare control
groups you find that there is there is no difference on IQ. Is it due to short term memory
problems? No, because again when you compare
with controls you have exactly the same results. So is it due to a deficit
in basic numerical abilities? Yes, it is. It is a stand alone issue
of this deficit and it is focused
on number sense, pure number ability, and we find that those who are
dyscalculic and those that are dyscalculic
and dyslexic are worse on enumeration
and number comparison, and for video one you will
remember that those are the core number abilities
which characterise our number sense. So what are the surface symptoms
of dyscalculia then? Well, the first thing is a delay
in counting, and what we mean by
this is when children are asked to start or end
at a specific point. So if they are given this array
and they say count the animals, ending with the mouse, they have extreme difficulty
with this because they have learned the procedure
of counting going from left to right, and they find it very hard
to to count in any other order. The use of less sophisticated
addition counting strategies. I’ve already talked about
those counting on and counting all comparing
those and also that more advanced ones starting
with a larger number. So I’ve talked about
this earlier, counting all, where you count
all the numbers, counting on, where you start with the
first addend, and then counting en masse,
where you start with the larger, but that does depend on knowing
what the larger number is, which for dyscalculics
is usually not the case. They also have trouble
memorizing number facts and we suggest that this
is because of a lack of conceptual understanding, it’s learning of tables is not
simply reciting them. You do have to have some
kind of concept conceptual picture of number to do it, and of course difficulty
with story problems, but that’s especially
where dyslexia is present. Now some of the problems
that have occurred and you remember
in the earlier video I talked about the fact
that we don’t use cutoffs, and the reason we don’t
use cutoffs is characterized on this slide. You can see here that there
are many tests for maths learning disabilities
and for dyscalculia, but you can also see that the
criterion that is set for being dyscalculic or having
maths learning difficulties is very different
from test to test. So this is from
a Butterworth article, and there are many reasons
for students having maths difficulties and it looks
like the standard tests are actually confounding
these difficulties. So we need to properly
characterize dyscalculia, not using these kind
of average normal tests. If it has a genetic cause
then we suggest it may well given that it is there in birth
then it’s likely to affect the basic numerical capacities
and the tests need to focus on these. Often these other tests don’t
focus on core number abilities, they focus on the acquired
or the cultural number abilities. So in our Melbourne
longitudinal study then we were looking to see
whether or not we could find indicators of dyscalculia
in terms of these core number abilities and we traced
the evolution of maths ability in two hundred
and sixty seven students starting at prep going
all the way through seven years through to grade six, and there is an article
school reference there for you on where
this is published and the first thing we measured
was basic number competency. This is core number
ability that I’ve talked about several times. So:
enumeration, estimation, non-verbal computation,
and counting, these sorts of skills really
are part of our basic toolkit for progressing into any
other arithmetic processing, and then we look to these
applied number competencies. So reading and writing
numbers and single digit addition. We need to know not only
can we characterise basic problems in number
but do they have an impact? If they don’t have an impact
then it doesn’t matter so much, but if they are impacting
students future predicting their arithmetic performance
or reading and writing numbers and single digit
addition then is an important
aspect of their school lives that we need to address, and the other thing we needed
to do was to look at just general
cognitive abilities. We need to know that this
problem dyscalculia, isn’t just some kind of a
general cognitive problem: a problem with IQ, a problem
with working memory, a problem with basic reaction
time or executive function. So remember in video one
I did say what subitising was but I reiterate it here
because it comes critical in going through this data, when children and adults
are asked to identify the number of dots in a visual
array they typically enumerate small arrays those
with less than or equal to four dots rapidly
and accurately and are slower and more
error prone in enumerating the larger arrays, the apparent automaticity
with which we can enumerate small dot arrays
has been termed subitsing. So it does enumeration
then we would present one to eight dots randomly
and this is the sort of profile that we get.
Now the y axis here, the vertical axis,
is a time measure. Mostly with dot counting
students can count them and get the right answer. What we’re interested in is how
long do they take? You can see here
that this function starts quite flat and then goes
into a greater angle. Now I just want to caution
about what happens at 7 and 8 it looks as though they’re
subitising again at 7 and 8. Students quite quickly learn
when you’re presenting lots of repeats of these
dots that the largest number of dots is eight, and so what they do is after
a while they start to answer very quickly for eight
just because they can see it’s a large number of dots. So you need to really consider
that graph after 7 and in fact some of our studies
now we present dots up to nine so that they can cut
off nine and get the accurate function and then we have
symbolic number comparison where we’re looking at
the numerals 1 to 9 and we’re asking students to say which is
larger and note again the amount of variability
that we’re finding in this profile. Now again it’s reaction time
and you can see that they’re getting slower
as the ratio increases and I will describe in video
four how we calculate this ratio. Essentially the higher
the ratio, the closer together
the two numbers are. So they get slower
as the numbers get closer together, but we have a great
deal of variability. So what we need to look at now
are these individual differences. We think that these
individual differences are going to give us
information about delay, deficit, and the typical
function and that is why we need to look at these
individual differences, not for a cutoff, where we might think
of dyscalculia as a “you either have it or you don’t”. It’s very much we want
to look at profiles of performance so we can see
the nature of the learning difficulties. So we’re suggesting then that
there is a real significance in the individual
differences in terms of indexing learning
difficulties like dyscalculia and also
different developmental pathways. So it’s very informative
for us in how students develop their maths abilities. So let’s briefly return
to this issue of performance variability and those
are the two graphs with dot enumeration
and symbolic number comparison. What we’re interested
in is whether or not that variability is showing
a systematic different groups. So, for instance, is there a group that’s at
the bottom of that range all the way, so the fastest? And is there another
group that’s at the top, the slowest? Or is it that in fact
is just randomly distributed variability
and doesn’t have any systematicity at all? And similarly
with number comparison. So why are we interested
in dot enumeration and number comparison though? Of course the reason we’re
interested in it is because it has been
shown to be a core number ability but very little
research is actually investigated whether
the failure to subitise or to subitise slowly
or to subitise to a very narrow span is associated
with other deficiency later, and other number deficiencies. So we are strongly advocating
that this is a diagnostic way of finding out about
and identifying dyscalculia, and so what we were interested
in was were there differences in subtitling
ability in five to six year olds and were these
in any way systematic? That is to say some students
were similar to each other in subitising ability
but different from other groups of students
in subitising ability, and if there are these
distinct groups, what is the relationship
between these differences and performance on cognitive
and maths tasks and is subitising speed related
to maths and cognitive performance? So in young children
then there appear to be variation in subitising speed.
We’ve already shown that, and we also showed in video
one that in preschoolers, we could distinguish
between groups. So how do we identify
speed profiles? Well, we actually use a new… a different sort
of analytical technique called latent class analysis, where latent class analysis
can identify inner sets of data, individuals who are like each
other but different from other individuals, who in
turn are like each other. So we identified these three
different profiles and you can see here that there
is the length of time is on the the upright y axis
but you can also see that this looks very much
like our variability graph where the top of the bars
in the variability graph have come out to be a single
group of like students. You can also see
from that that those students are not subitising, they are counting pretty
much from the beginning. Some is are subitising from two
in that group but most are not subitising at all, and you can see that in the aqua
colored group that they are
subitising probably to four for most of that group,
and then we’ve got the, what you might describe as the
average group, in between those five to six
year olds we would normally expect them to be able
to subitisse to three and that middle group
are able to do that but it’s important
to note that the profiles didn’t differ in terms
of the age of the students. It wasn’t just that that top
rate group were older or younger than the other students. If they didn’t different digits
span so it wasn’t an issue to do with working memory
or short term memory and they didn’t differ
in visual spatial working memory either that is just
a memory for how blocks are touched and they importantly
didn’t differ in reading comprehension ability. So these groups differ
specifically in number and they differ in a particular
way with respect to their subitising. Now if you look at
the part of the graph which is the counting range, so if you look from four
to seven there, you will see that the speed
of counting is the same for each, the rate of the speed
of counting is the same, the groups differ in terms
of overall time but the actual rate
of getting from one number to another is different, but it is not like that
in the subitising range and then we looked at
reading and writing numbers as a function of these groups
that we determined in prep and we went to look to see
what happened in Grades 1 and 2. Well the first thing
that happened in year one you could see that the slow
group were only about 28 percent successful
on reading three digit numbers. This graph tells us two things. It tells us that the groups
did differ on their ability to read three digit numbers, both at year one
and it year two, but you can see the progress
that is made by the slow group from year one to year two. So we need to keep that in mind
when we’re worrying about students
with dyscalculia not making progress.
They do make progress. The issue is whether they’re
making progress in the same way as typically developing
students and you can see that they’re not too different
from the average student in that second year.
Similarly with witing numbers, again though performance
generally on writing numbers is lower in years one
or two and you can see again that there were differences. It’s also important
to note that there were differences in single
digit addition and also four years later, because remember this was
a developmental longitudinal study, four years later
using multi digit, so three digit addition,
subtraction, multiplication. The student groups did
differ at that point. So there was a great deal
of impact of this subitising ability on performance later. Now do dot enumeration
differences extend into adulthood or is this just
something that happens in school? We were interested in this so we
took 88 adults, they were aged from 18
to 61 and we just applied the same dot enumeration test, counting dots up to 8 and what
we found was almost exactly the same. So yes,
it does extend into adults. Adults are much faster
overall but we had these distinct groups and you can
see there in the blue group, they are not subitising
so this prevalence rate does extend to adulthood and of
course in the video we saw an individual
adult with subitising difficulty and problems.
Now I want to mention Lucy, who is part of this “all
in the mind” program. Lucy was referred to us
a number of years ago now because her teachers said
she really was not processing number at all
well and they didn’t understand what was happening
with Lucy. So we tested Lucy using our
test that we’ve devised from this longitudinal study, and Lucy probably has one
of the purest forms of dyscalculia we’ve
ever witnessed. On dot enumeration, she had a linear increasing
function she had no subitising whatever
she counted dots at two and Lucy also show her poor
performance on other indicators or other indicators
of basic core number ability, like number comparison task
although she was average and above average in working
memory and processing speed, but importantly Lucy was an
extremely able student. She was fluent
in several languages. Her two parents were speaking
two different languages French and Italian. She spoke fluently in French
to one of her parents, fluently in Italian
to the other, fluently English in English
to me and so she had her school
performance generally was very high but she
had this specific difficulty with number
processing and I worked with Lucy for a while and we
were working particularly on equivalent
fractions which were mentioned, we talked about fractions
in video 2, and we were working with flip
cards and for a considerable amount of time Lucy
had no idea what to do with these flip cards. Eventually though in one
session she suddenly had this breakthrough and she
suddenly understood all about equivalent fractions
and she could work with these flip cards and then she was able
to work on the whiteboard and when
her parents came to pick her up we were
so excited to show her parents how well she had
suddenly got the idea of equivalent fractions
and I was feeling very relieved that we would really
overcome this hurdle, and Lucy was going
to be able to move on. When Lucy came back for a
session the following week it was as if she had never
heard of equivalent fractions. The reason for that is likely
that she suddenly realized what the process was in order
to do equivalent fractions, but because she didn’t have
a conceptual understanding of what that process was about, she didn’t then manage
to remember it. A week later when she’d
had a whole week of her life and the exciting
things that happen for nine year olds and her lack
of number sense seemed to result in her not
having any sense of the relationship
between numbers. So she didn’t know that 30
was less than 40, and when I asked her once:
how would you get from 37 to 38, what would you need to add?
She had no idea. She she was frustrating
in some sense but exciting in others because she did make
progress and that was exciting, but if you said to her what do
you picture when you when I say
to you twenty seven and she says I picture a two
and a seven and that is exactly how
students with dyscalculia are imagining numbers. They’re not imagining
it in relation to the understanding
of number that we need, so this then leads us directly
into assessing students for specific number difficulties
and in particular for dyscalculia: the focus
of video four, but before we do that let’s
just summarise what’s happened in videos 1 to 3.
We’ve highlighted a wide range of maths abilities
that students are expected to acquire through
formal schooling. The capabilities they had
entering school and the wide range
of difficulties that can occur along the way. We’ve drawn attention
to different developmental pathways and the importance
of recognizing the wide range of students’ maths performance.
In videos three, I’ve focused on a specific
type of maths learning difficulty:
dyscalculia, which is qualitatively
different from other maths learning difficulties. Dyscalculia has a neurological
underpinning and that impacts the complete
range of maths processing. In the final two videos
then we will cover ways to assess maths learning
difficulties and intervention strategies
to support MLD students in successfully
accessing the maths curriculum.

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