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.