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

about already. So how might we assess it? Well, first we might ask students

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

you immediately you were able to answer but when you had

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

abilities that we’ve talked about.

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

again very easy to administer, students read Arabic numbers

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.

So reading numbers, we talked about reading numbers

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

going to show you some numbers please read them to me

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

in reading numbers, and we will talk about

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

into tens of thousands they immediately jump to a

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

going to show you some adding sums, please tell me

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

addition. Retrieval is, I knew the answer that is to say

they now had that number fact. If you think back we talked

about math learning difficulties showing

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

nine” or if they’re asked for six add five they say

“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

and they’re asked to read them back and I’m not

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

arithmetic achievement test on addition

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

you information about this variability partitioning

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.