# Learning Difficulties in Numeracy – Classroom Assessments (4/5)

This video is the fourth of five
that have been developed in a collaboration between
the University of Melbourne and the Victorian Department
of Education and Training to examine maths learning
difficulties in students and especially looking
at dyscalculia. This video is looking
at assessment. How might we assess maths
learning difficulties so that we have a better
idea of what students are and aren’t able to do? That will inform appropriate
intervention strategies which we will address in Video
5. So the Department of Education’s website
suggests some useful maths assessment tools that can
be accessed by teachers. The maths online interview,
for example, contains some useful milestones
for early maths development and it maps common
responses to the curriculum and that’s an important part
of using assessment that it’s going to map
directly onto what the students need
to know and what they’re learning in schools. Now, there is a further source
of information that is provided through the NAPLAN results
and I’m not going to talk about the specifics
of the NAPLAN results, only that we are particularly
interested in the way a student’s performance might be
different in literacy and numeracy, so not
the absolute results, but if there is a real
discrepancy between the band for numeracy and they
are well below the band for that age level
in numeracy but they are either on the band
or above for literacy, we would use that as a flag. Its a flag that they might
be at risk of being unable to progress satisfactorily
in maths without targeted intervention and we
would probably think that they needed to be
assessed for dyscalculia. Now, other standardised
measures of MLD are are available. There’s a diagnostic
mathematical test for junior grades and each
test covers skills into two parts;
part A is pure number, under their categories:
enumeration, place value, counting, recalling number facts
for operations, number sentences and fractions.
Part B is applied number, length, time, money, space,
capacity and so on. Now thinking back
to the earlier videos as well you would look at
that and say “Well I’m not sure Part A is pure number. It includes also acquired
number abilities like number facts and the operations.
Then there is key maths revised. This comprises scales
in three areas: basic concepts enumeration
rational numbers and geometry, operations, applications
of measurement time and money and so on. Now you’ll notice there
rational numbers and geometry are being
categorized as basic concepts. Again, we wouldn’t
consider those to be basic concepts. So both of these tests
then are covering early arithmetic development and are
probably more useful as indicators of specific
difficulties a student is experiencing but the test
is not measuring basic number abilities as we’ve
discussed in videos one and three. Then there are two
other possible tests: progressive achievement tests
in mathematics measures knowledge for years level 3
to 5, 5 to 8, and 6 to 9. Again these are in the areas
of number computation, fractions, statistics,
measurement, spatial relations, relations and functions,
logic and sets, and then there’s an ACER Test
of Basic Skills, which has a sample number space
and measurement in upper primary levels. Number is assessed
by the ACER test through arithmetic operations, manipulations of counting
patterns, arranging numbers in order,
manipulating placed value, selecting the appropriate
operation, and word problems, but again all of these
are acquired number skills. They are things that students
are learning in school rather than core
number abilities. All of these would be
categorized by us as acquired or applied number competencies, not pure basic
number competencies. So which tests
should teachers use or should they use
any or all of these? Well we are more interested
perhaps in looking at the very basic processes that underlie
numerically relevant computations. So we’d like to separate number
understanding from calculation but we think of understanding
as this concept of numerosity — number sense — the basic number skills
and calculation is the operations
on those numbers, and we’re going to distinguish
between those two things, and we’re talking
about assessment. So we’re interested in using
tests of capacity rather than tests of attainment. If you think back to an earlier
video where we talked about capacities
and then measurements of those capacities which we regard
as attainment so the capacities we’re
interested in are core number abilities and the attainment
of the acquired number knowledge, the knowledge that students
learn in school as they progress through the maths curriculum. So we can use various
tests to assess aspects of numerical development.
We can use counting, dot enumeration,
magnitude comparison. We’ve talked about all
of these in in previous videos and we would describe
these as core number abilities and although counting is not
as pure as magnitude comparison, you aren’t able to do
dot enumeration without being able to count. So counting we were regard
as a core number ability and then we would have other
indicators of performance: reading numbers and single
digit additions and we would describe those
as acquired knowledge, they’re the cultural knowledge, the applied knowledge
of number that occur in the school curriculum. So we are going to want
to assess each of those, we need to know how students
can perform on core number abilities and we need to know
how they can perform on their applied
number competencies so that we can compare them. Is there an impact
of one on the other? And we’re going to start
with counting which is a very basic
fundamental aspect of number which we’ve talked
to count items in an array: just count these items for me,
please. Or we could say start
counting out loud and keep going until I say stop. That’s a measure of how
high the maximum number a student can count
accurately to. Or we might say give me
a set number of items, we talked about that in
the preschool tests that we ran that Sarah Gray
ran with Bob Reeve and then we need to know
how students approach counting when they’ve got
different arrays or arrangements. So if the items are in a circle
for example and then we have these
different starting points which I mentioned in the
dyscalculia video three: count the animals starting
with the fish. This is a known problem
for students with maths learning difficulties
that they have to inhibit their desire to just
count from left to right. So counting is a common
human activity, but how do counting
abilities develop? Well the first thing
and very critical is that the realization
that all sets of a certain number have something in common
and that is equality and that’s the both most
basic numerical understanding. Now counting though requires
a number of principles and these principles… so if you have a student
who when you you’ve run these assessments is not
able to count, what are the things
that you need to look for and what intervention
do you need? Well the first thing
a student needs to know is that count words
are in a fixed order. They are recited
in a fixed order, and that doesn’t change. So that order needs
to be learned. They also need to know about one
to one correspondence, that for every count
would you touch an object or you cross off
an object in order to count. They need to know that the order
of items counted doesn’t matter, as in that problem
with a different starting points. They also need to know
that it doesn’t matter what your counting, the manner of counting
remains the same. So you could be counting
anything at all and it will be exactly the same
with the fixed order, one to one correspondence, and it doesn’t matter where
you start, and then they need
to know that the final count word represents
the numerosity of a set of objects. Now this was raised
by Trish, learning support teacher
in the video clip in video three where she talked
about how important it is to understand
that when you get to the number five, that also includes the other
four objects in the set. Now these counting principles
were articulated by Professor Rochelle Gelman,
she was a psychology, a very well known psychology
professor who specializes in children’s understanding
of number. Now Rochelle will argue
though that basic number skills may be a natural
and universal human ability that we’re predisposed
to acquire. So they the desire to count
seems to be very early and continues on until
competency is reached. Now most but not all three
year olds can count to 10. Most five year olds can’t count
to 30 and above. Once they’ve learned
the regularity and the count sequence above 20, counting becomes much
easier once you got there, and not until the age
of five do most children know the relative sizes of the
numbers between 1 and 10. So they aren’t able to say,
without seeing the objects, which has more four
or six oranges, and this is to do
with numerical ordering, but it’s the mental picture
of numerical ordering, but there are some significant
cultural differences in counting level and I’ll
show you this graph and you can see if we look at
three year olds in the highest number
counted in the case of the United States students. Ten is the average
highest number counted but for the Chinese children
it’s rather higher than that, but look what happens at
four by four the Chinese children have moved
on considerably, those in the United States
haven’t yet, and this is this issue
of the numbers from 10 to 20 being irregular. Different count words and they
have to all be learned. So to some extent then there
are cultural differences between students and they’re
mostly attributable to language so children
start to count early, as early as 18 months, and at first they may use
some sort of idiosyncratic sequence 1 2 7 9 3 4,
but they do acquire the number words. It is of course in some
ways that the number words are somewhat arbitrary, so they have to be learned
by rote in order and they may have a fragile competence
on that count sequence for quite a long time
but they are able to judge whether puppets
counting of objects is correct. So that would suggest
that they can recognize when the numbers
are in order but not necessarily generate
them for themselves when they’re very young, and they often start
as something we know as “one knowers” so they know
that they object one they understand
the idea of a one. So if you ask for one
goldfish they’ll give you exactly one goldfish
but if you ask for two three or seven they’ll
give you a handful of fish. Never one though, but they won’t reliably
give you more for 7 than 2, and if you ask a one knower: how many fish on this card
and the cards have one to eight fish, they’ll tell you one
for one fish and two fish for everything else. So this is part of the progress
into counting, we’re talking about very young
children here. Now dot enumeration, we have talked about
in the previous videos and I just want to say
as I’m going through these core number abilities that we
are not anticipating that the teachers will makes
the stimuli and run these tests. I’m talking about them
because they are measures of core number ability
and they are always included in some sort of a
screener for dyscalculia, and we’ll talk about
a potential version of a screener later but it
is important to know what is determined,
what is assessed, in dot enumeration,
and why it’s important. So students are asked
to identify the total number of dots in an array
so you will say I’m going to show you some
pictures of dots, tell me as fast as you can
without making mistakes, how many dots you see, and then you record the response
and the response time and again that response
is usually pretty accurate but the time they take
is the fundamentally important discriminating issue. So this is what the arrays
might look like and so I’m going to get
you to try this test now, this is an example of what
we would present to students, a very
small example. So I just need you to call out
how many dots there are in these arrays. So what did you notice
while you were doing that? This is the typical plot
of dots enumeration, and I suspect you notice
that when you had small arrays like three and two
the larger arrays you had to stop
and individually count the dots, and that’s exactly what happens
in dots enumeration. So you can see here
that we have more or less flat line up to three
and sometimes up to four. These were five and a half
year old students. So that’s why three
is probably their maximum subitising with some
subitising to four. If it was adults you would see
that probably to four so there are differences
in the subitising and the counting ranges,
and that is really important. That difference between
the simultaneous enumeration versus counting
is really important, and of course here we’ve got
lots of error and those error bars we’ve talked about
at some length when we talked about
partitioning the data in video three. Now maybe due comparison
is the other one of these core number
So dot enumeration, magnitude comparison
are really accurate measures of core number. So we would compare two
numbers or two quantities and indicate which is larger, and these can be presented
either in a non symbolic form or in a symbolic form.
Non symbolic: students compare
two sets of dots, squares or other items
and identify the set with a larger number of squares
and they might look like this. In the symbolic form students
compared two numerals and identify the larger
in magnitude, and it would look like that. Now in non symbolic number
comparison we use all the comparisons except
for doubles of course of the numbers one to nine
and the students are asked to decide which side
has more squares. Responses are recorded
for accuracy and response time, the stimuli are coded
according to the ratio of the number of squares
and I mentioned this earlier in video 3 when we
were looking at the number comparison
function and I said that as the numbers get
closer together the students get slower
and that’s to do with this ratio. So essentially the ratio
is calculated by taking the smaller number
over the larger number. So in the case of five on one
side and six on the other, the ratio is 0.835 divided by 6
and we call that ratio eight. Three divided by two, take the smaller of the larger
point 0.67 ratio 6. So here’s an example
of a ratio 2 and that’s 2 on one side and 8 on the other. Now why are the squares
of different sizes? Why have we done that? And the answer to that is that
we need to control for total
surface area. If you’re presenting these
to students and they’re able to use the amount of blue
to make that determination, then we are not getting
an accurate measure of their number ability. So we’re careful to make
sure that the sum of the area, the amount of blue is identical
on both sides of the display. So instructions for non
symbolic number comparison will be something like this. In a moment you’ll see
a picture of squares, one side of the picture
will always have more squares than the other. If you think this side
and you indicate the left has more press the yellow dot
and if you think the other side, press the red dot. So it
would look like this. This is that so in all of the
dyscalculia screeners and any measurement that we use
for measuring core number ability, this is the sort of thing
that we would present and that’s ratio 2,
and then I’m going to include a ratio 8 so you can
see how much longer it takes to decide which side
has more squares and that’s a very good
indicator of of your discriminate ability
for the numbers. In the symbolic form
it would look like this, and they again discriminatable
and as they get closer together the students get
slower. Now number line, we’ve talked about number
line quite a lot but it is a really simple task
that can be conducted in a classroom. If you want to know how well
your students are mapping a symbolic number
onto a spatial array, this is a very simple task
for you to do and all you need to do is to have
an A4 page or a set of A4 pages onto which is ruled a 25
centimetre number line and a target number
at the top of the page. The beginning of each number
line is marked with zero. The endpoint mark with a
thousand or one hundred if it’s for very young children.
So it would look like this, and so for prep to grade
three we would probably look at these and probably
starting at Grade Two would start to introduce
the 1000 line and that would look like that. Now students vary in their
response profile according to the degree that they have
this accurate mapping of symbols onto
the spatial relations of number. So if they have this ordinal
conception of numbers which tend to overestimate
smaller numbers and underestimate larger
numbers that I talked about earlier especially in students
with maths learning difficulties, we would get this kind
of a profile and we’d say this is a logarithmic profile, that is to say it is at
the start of the graph. We’re overestimating all
the small numbers and then you can see
that the larger numbers are significantly
underestimated. Students who understand
the relationships between numbers of different magnitudes
will accurately map the numbers in a linear
progression and it will look like that. That is very informative
and quite easy to administer, and so what you would do
is have say 20 numbers that you asked the students
to map and you would do it as a classroom exercise
and then measure the difference between
where these numbers should be and their accuracy. Now we also mentioned though
that in order to do the number line estimation task you need
to be able to name the numbers and we call this number access: the ability to access
looking at a numeral what the word is for
that numeral and this can be an impediment
if students aren’t able to do this because they can’t do
some of the other number tasks like symbolic number comparison
and like the number line task. This is a very simple task
from 1 to 9 randomly presented on a computer screen. So each number is presented
three times just in case there’s some kind of a
distraction or one of them because what you’re
interested in here is the time it takes and their accuracy. By the end of grade 1
you would expect them to be accurate. So then you’re interested
in the time they take because it may be that they’re
slower to access the word for the number. So you’re interested
in accuracy and response time and accuracy is often high. This is a sample
of the stimulate, and what we also do though
is we for comparison we do check this symbol access
on letters as well because we’re interested
in whether or not there is a different time taken
to access the same symbols for a number than the letters.
a few times in these slides already. So the sorts of things
we would do is we’d have 36 trials six of each length again. We always have a number
of repeats to make sure that there isn’t just
an individual error happening here,
but it’s more systematic. So we’d have single digit teen
numbers two three four or five digits and we’d have longer
strings for year five up which you might go into the
hundred thousands and possibly the millions. So the instruction then is I’m
and you record the actual response the accuracy
and the error type. So these are some samples
of what you might present and you can see the range
of the different string lengths there. Now the reason for administering
a reading numbers test is to look at
the errors that the students are making because it’s
the errors that are informative. If you think back to video
to where we talked about writing numbers and how
important it this to understand place
value and to know how the numbers fit into those
plans values. This is also the case
writing numbers shortly. So this is an example
of the sorts of errors that may occur. So if we look at substitution
that is there presented with one hundred sixty
two and they say one hundred and seventy two. Now if that’s a one off
then it’s likely to just be a slip of the tongue, but if they’re systematically
doing that then we would say there is an issue here to do
with their substituting numbers and possibly their symbol axis. Transitional errors using
the wrong starting number, the wrong multiplicant, so for instance 2 3 0 1
being read as two million three hundred and one, and in fact when we’re doing
our testing when we get into five digit numbers
these errors start to come in quite predominantly. We find that students are okay
up to the thousands and then when they get
million and then we’ve talked about these placement
values reading 1002 as 102, that’s seeing
that hundred separately without the correct place value. If they read the digits
as if they were just not part of a single number 2 3
0 1. That would suggest to you that they are not
understanding it as a number at all. They’re just seeing it as
a string of digits, a little bit like Lucy did
when I said just picture what 27 looks like and she’ll
say 2 7 and then reversal these last
two errors may well just be some independent errors
that come in, but if they’re
systematically making reversals or say 19 is 90
or 60 is 16 then those need to be pursued to check
to see whether or not there is a basic issue there
with their reading number ability. Writing numbers is the other
side of this transcoding coin, and again we’re going to have
examples of teen two three four or five digit numbers
and they write the number on a sheet of paper and in fact
that earlier sheet of paper which I showed in video 2
is an example of these being written during a test
of a group of students, and we’re interested again
in accuracy and error type. So these are the example
numbers 2 and 7 are usually the practice ones
and then they write the teen numbers
and so on and there are similar errors as occur
in reading numbers. Now they might just
write a digit incorrectly and it might
be a slip. So it’s important just
to check every time they’ve done a three digit number, have they made that same mistake
when they’ve done larger numbers? Are they making mistakes
with substituting numbers? Important areas are things
like 2005 being written as two hundred and five or two
and three zeros and then a five, ninety eight is eighty
nine again. These might be just
independent errors that are happening but if
they are systematic then they mean something, and the concatenation error
that we’ve talked about earlier in video 2 where
they are not putting the right place value on the numbers, they’re just writing them
as if they’re each part of the number. Single digit addition
was mentioned in video 3 in the longitudinal study along
with reading or writing numbers as an important outcome measure. We need to know how
students are performing on these other tasks
as a function of how they understand number. Single digit addition
for 5 and 6 year olds, we would have all
the combinations from 2 to 7. So the reason for that is so
that the solution is a single digit, so for young children
we like to keep within the single digit range. Older students then we do all
the combinations from 1 to 9 and the instruction is I’m
the answer, and when the student
answers you say “How did you work that out?” and that is the important
measure from this test. What strategy is the student
using in order to perform addition? Think back to earlier videos
where we’ve talked about sophistication of counting
skills whether they’re counting all,
counting on, counting on from the larger, all of those are important steps
in the sophistication of understanding number in the
service of these operations. So we record the student’s
answer and the counting support. Did they use their fingers
to assist them? This is an important indicator. Often it’s it’s a positive
when they are using their fingers because they are more
likely to be accurate but it is important
to note it and whether they’re also mentally, you can hear them quietly
reciting the words and then you’re interested
in what problem solving strategy they used
and how long they took. It’s worth noting though
that certain certain strategies have longer time, so counting all is bound
to also have a longer time, and this is what the stimuli
might look like. Four plus three, two plus seven, so the strategies
then are the things that are of great
interest to us, some students don’t have any
strategy at all. So I have tested students where
they’ll say I don’t know sometimes they’ll say
it’s my mother’s birthday today and she’s 33.
So I said 33. So these are answers
that don’t seem to relate to the numbers
that are being added together, and then we have the count
all strategy that was described in video 2
counting on from one and starting and counting
all of them, counting on where we actually
start at the first addend and then count on the remainder. The min strategy where we’re
selecting the larger of the two numbers and counting on, and then we come to two
quite sophisticated strategies in single digit
they now had that number fact. If you think back we talked
that these sorts of facts were hard to retrieve and that’s
because they don’t have a conceptual understanding
for them so they can’t draw them from their memory. So when they are saying
they knew the answer and they’ve got it right
then they retrieved it from their memory. Decomposition is sophisticated
because the student is using an answer
from another problem that they’ve done which is
related in order to get the answer to this problem. So for example for two
plus seven they will say something like “Well I know
that three plus seven is ten. So two plus there must be
“Well I know that two sixes are 12. So this must be 11” and those
are sophisticated strategies that students with normal
expected progression will have and students with maths
learning difficulties will rarely have
because it depends on understanding the concept
of the number. Now we also mustn’t leave
the idea that number may be just a general capacity. So it’s a working memory or an
IQ capacity or something like that. Now there are still researchers
who argue that the differences in numerical competence arise
from differences in general cognitive abilities
and that there isn’t a specificity
for a number. So we need to use
standardized measures of cognitive abilities
such as non-verbal IQ, visual spatial working memory,
verbal working memory, basic language ability,
and processing speed. We need to look at all of those
in case number abilities that we’re picking up
are actually just within these cognitive areas. Now these sorts of tests
are usually performed by an educational psychologist. So we would not necessarily
suggest that teachers would conduct these tests
but if you have a student who on the basis of the other
things we’ve mentioned looks like they have dyscalculia
then it’s probably a good idea to get this ruled out
because it may be that it is a basic capacity
that is the issue. So the sort of measures
then of general cognitive ability would be non-verbal IQ. So there’s something
like the Ravens color Progressive Matrices. The reason it needs to be
a non-verbal IQ is we need to exclude all the possible
learned capacities and just take something
which is a pure measure, thought to be a pure
measure of IQ, and then visual spatial
working memory is measured with something called Cosi
block span which I will describe in a bit
more detail in a moment. Verbal working memory
is usually measured via the backward digit span. Now we’ve told you that working
memory and what why it’s useful in number when we’re doing
fairly complex calculations and we’re holding
particular processes and numbers in our
heads to do other processing and computations, that requires working
memory and the backwards digits span is simply, there are increasing strings
of digits repeated and the student has to repeat
them back in reverse order. So they’ve got to hold
them in their mind and then repeat them backwards, and that is exactly what’s
going on when you are working with numbers in problem solving
and then basic language ability, something like non word
repetition is an aspect of basic language ability and I
can describe that in some more detail. Processing speed is usually
measured by some form of basic reaction time. That means responding
as quickly as possible to stimuli presented
in different time intervals. So the reason for that is
to prevent anticipation, if they know it’s going
to come in a certain time interval they’ll
press in anticipation. So we vary the time
interval so that we can stop them from anticipation. So this is an example of what
the Raven’s looks like. You have a series of patterns
and each one’s got a piece missing and the students
are given six possible options for filling
out the space. So what they need to do
then is to look at the relationships. Now we have talked earlier
about the idea of analogies. They need to see the analogy, they need to see what is
the relationship between the left hand top, left hand corner with the ones
that are known to fill in that last square. Quasi block tapping task
I have talked to a few times now about visual
spatial working memory without showing you how
this is measured, so I’ll just show you here
how it’s measured. Essentially it has this setup. Now you can see that the front
here says the examiners view and you can
see numbers on the blocks. The examiner can
see these numbers. The student can’t, and what the examiner
does is touch the blocks. So we start off with let’s say
2 so it’s 8 9 and then examiners will
touch the block 8 and then 9 and the student
then has to touch the block 8 and 9, and the
reason for the numbers is so that the same task will be
administered for every student. So by having a set pattern
of the increasing string lengths using numbers for the
examiner the student is unaware of that although
sometimes when I’m putting the task away a student will
see the numbers and say so that’s how you were able
to do that, but essentially you stop
as soon as the student can’t manage two of one string length. Non-word repetition is important
because it’s not real words in the language
but they could be. So shrimp for example
isn’t an English word but it could be an English
word and you ask the student to read it back to you. So these
are presented orally, the students hear the words
going to dwell on all of these, you will be able to look
at these in your own time, you can see that some
of them are really quite long though and quite complex
and students have to… and they could all be
English words they actually aren’t
any of them English words but they could be. There are some commercial
products on the market for assessing number
difficulties and it’s important to say
that the Department doesn’t preference any of these
commercial products. We’re just presenting some
information about them and some of them we use in our
clinical assessments but there is the numeracy learning
progression on the VCAA website which can also just tell
you the various capacities that need to be addressed
when you are doing assessments and the team
of three is a test which aligns very well with this learning
progression for basic numeracy and as you can
see when you look at the learning progression guide, numeracy difficulty
increases within and across levels. So in increasing difficulty
and the skills are cumulative over time, so the team of three is able
to present increasing difficulty items
that the student can attempt and they are all
cumulative in terms of skills. So it starts and it focuses
on numbering skills, number comparison,
numerical literacy, mastery of number facts,
calculation skills, understanding maths concepts, and all of these
are considered to be foundational math skills
but again these aren’t core number abilities, they acquired number
abilities and importantly with this test although
it is norm reference then you can get a score
and find the the age level, the equivalent age level. We have a tendency to use
the individual items to tell us exactly
what the student can and can’t do. So we can norm reference
it and say well they’re working at the age of an average
eight year old, the test is designed from three
to nine but if you have an older student aged 14 who has severe
maths learning difficulties you can use this to find
two things. One, what their equivalent age level
is in terms of a norm reference but also what particular
strengths and weaknesses do they have. When we talk about intervention
in video five, their strengths and weaknesses
are exactly what we need to know in order
to have the optimum intervention strategy for them. So the purposes of the team
then is to identify the students who are
significantly behind or ahead of their peers
in the development of math thinking, to identify specific
strengths and weaknesses in the mathematical thinking
using individual items rather than just as an average test, and also to suggest
instructional practices that might be appropriate
for individual students. If you pick up a particular
weakness you can intervene on that specific weakness
and you can also document the progress that they’re making
in learning arithmetic along the way. There’s also the dyscalculia
screener which again we use in our practice and these
tests were selected as the most effective in discriminating
students with dyscalculia because it
focuses on core number ability, this is not about acquired
number at all. It’s very much about core, although there is an acquired
number test there, and that is to look at
the impact of the core number ability on that outcome measure. These involve comparing
numbers and counting dots, as we’ve mentioned earlier
in this presentation and also in video one. The screeners specifically
designed to identify students with basic
number deficits, it’s not intended to predict
a student’s future maths attainment or for example
on geometry, algebra, or topology. It’s simply how how much basic
number knowledge do they have? What is there number sense? So the screener measures
first of all basic reaction time, the reason for that is the other
measures are timed measures so you need to know
that if they are slow on subitising or slow
on number comparison, it isn’t that they are overall
reacting slowly. It is that they have these
specific number deficits. It includes dot enumeration
and number comparison, R2, core number ability tests, and then there is this
and multiplication which will give a measure
of the outcome for a student with this particular pattern
of number sense or number ability. So the starting point
for developing the screener was to evaluate a variety
of tests all designed to understand numeracy across
a huge range of age groups from 6 year olds to adults
and then case studies of individual students
and adults with MLD showed that those with limited
capacities as measured by performance on these tests
of number comparison and dot counting almost invariably
had profound difficulties at school in learning mathematics, so they were clearly good
indicators of dyscalculia. So that brings us back
to the discussion of performance variability
and the importance of identifying
individual difference in students
numerical processing, so we’ve claimed already
that there’s a huge variability in maths
understanding even in the young, and this variability suggests
that there are different patterns of maths understanding
and these patterns of misunderstanding
indicate different developmental pathways
and different developmental pathways may
have different long term outcomes, and to do this we need
a different form of analysis. It isn’t sufficient just
to take an average of the performance of a group
of students and think that that is going to give
the variability is important as we described in video 3
and if we want to make claims about the predictive nature
of these understandings we need longitudinal data preferably
over a long period and of course students
are in school longitudinally across a number
of grades. These tests can be repeated
and you can see how they can progress across the years
and you can build a profile of their
development in maths across that, and we need to continue
to challenge this age as a proxy for a development
approach to understanding. Students in one grade
will vary enormously in their maths ability. So we can’t say this is the age
you will be doing this because it will be
a huge variety within a single grade, and that’s why we look at
this different approach of profiles,
individual profiles. So where do we go to now then? Well we need to identify
the origins of competence, we need to assess core number, acquired number and cognitive
abilities to establish the student’s basic
number understanding. We need to partition
this performance to identify individual differences
in number ability and that can be done
by administering individual tests to highlight
the differences, the strengths and weaknesses. We need to note that not all
children learn in the same way, and teaching will be tailored
to the individual abilities and the needs and then
we need to intervene. We have all
this information now, we have the information
about individual students, what their strengths
and weaknesses are. So now we need to tailor
intervention to assist them in accessing the maths
curriculum and video 5 is going to address intervention
strategies for students with maths learning difficulty,
especially dyscalculia.