Welcome to the first of five videos that have been developed in a collaboration between the University of Melbourne and the Department of Education and Training. In June 2017, the Department of Education released its literacy and numeracy Phase 2 initiative which outlines Victoria's plan to improve literacy and numeracy for all students. Phase 2 of the strategy released in June 2018 includes a commitment to deliver numeracy teaching resources and guidance for teaching students with maths learning difficulties including dyscalculia. The Department of Education commissioned this series of videos for the purpose of developing the understanding of teachers and school leaders about learning difficulties in numeracy and dyscalculia. These five videos have been designed to provide teachers with useful and usable information about how to best support students with learning difficulties in numeracy and including dyscalculia. I'm Dr. Judy Humberstone and first I'd like to introduce the team. Dr. Brian Butterworth and Associate Professor Bob Reeve have been studying maths learning difficulties from clinical and educational neuroscience perspectives for over 30 years. Brian wrote the highly influential book “The Mathematical Brain” and more recently Dyscalculia: From science to education”. He is responsible for setting up one of the first dyscalculia clinics in London prior to joining the University of Melbourne. Bob was at the University of Illinois for eight years and was involved in educational assessment programs designed to facilitate maths skills for children with maths learning difficulties in Chicago and Oakland, California. Brian and Bob have published a large number of research papers and chapters on the nature of maths learning difficulties and frequently discuss their work in national and international meetings. Dr. Sarah Gray and myself are involved in the assessment of dyscalculia and mass difficulties more generally at the University of Melbourne's dyscalculia clinic. I'm a former Victorian Department of Education school principal with 23 years experience of teaching maths at a secondary level. I'm a registered psychologist and was jointly responsible for setting up the dyscalculia clinic at the university. I provide advice to schools and parents on how best to help students with mass learning difficulties and dyscalculia and I also present at national and international meetings. Sarah is a clinical psychologist who works in the dyscalculia clinic at the University and also with educational groups. She is particularly interested in how to provide students with maths learning difficulties with emotional supports. She has published a number of well regarded papers on this topic. It's also important to note though there's a large number of high degree and postdoctoral students who have contributed to our understanding of maths learning difficulties. We'd particularly like to highlight the contribution of Dr Fiona Reynolds who has been involved in this work for many years. So let's start in the classroom. We know that teachers are great clinicians. They are the people in touch with students every day, and they have a view of all the sorts of math learning difficulties that occur and the wide range, and these are the sorts of things that teachers report to us. They report this wide range of maths performance within a single year level, and that will extend to very high performing students who are well above the benchmarks for their year level. Those who are performing at an average level are the majority of students and they acquire skills appropriate for their level. Then we have a group of lower performing students with some maths learning difficulties and they require additional help. A fourth group of students also with massive learning difficulties exists who seem not to retain or transfer that knowledge. So how might we characterize these teacher observations? Well, the average performing students we might regard as expected maths development. These are the students who are following the curriculum benchmarks in the order that they would be expected to at the appropriate level, and of course there are some students who are going to be advanced and they will be also following these standard curriculum. These lower performing students with MLD require additional help, and we would describe these as delayed because with that additional help often these students, they may not catch up, but they certainly make the right degree of progress over time, and then we have this fourth group with MLD who seem not to be able to retain or transfer their knowledge and these students have a neurological deficit which is impacting their development and this will be unpacked during the course of these videos more explicitly and in video 3 in particular. So our aim then is to understand what these individual differences are and some of the members of our group have been studying MLD for over 30 years and our understanding of basic maths development is well behind our understanding of language development but this is something that we are addressing by producing these videos and also research has has made significant progress over years, which I will be talking about. So, there are all kinds of reasons for students finding maths problematic, essentially maths is the componential system. That means we have increments, one builds on the other and these performance differences that we've described earlier might result from missing class because of illness, ill attention, some socio economic factors, some anxiety about maths and dyscalculia and all of these factors and more need to be taken into account when we're determining the nature of students maths difficulties. Given these individual differences in maths learning we consider it useful to make some distinctions between types of developmental change and we are going to talk about those that are on a typical developmental pathway and those will include the advanced students who are still typical in terms of their progress and then we have the set of students who are delayed, who need intervention and usually make good progress with intervention, and then we have this group of students whose development is impacted by a specific deficit, who sometimes are not making as much progress with intervention and we're going to suggest through these videos ways we might assist those students and provide some interventions for them which may have an impact on their development and the important thing to take from this is that if you take an average learning progression you are not going to pick out these different sorts of development and change. So we do need a different approach in order to to isolate these different sorts of development so we can have appropriate assessment and intervention and we need to better understand these individual differences in children’s maths abilities which are picked up by teachers on a day to day basis, and what we are doing I hope is simply articulating them and making some suggestions as to how we might address these differences, and of course within these developmental progressions some students may follow a different trajectory but nevertheless we're going to continue with distinguishing in this way and pursuing these different sorts of developments through these videos. Now maths learning difficulties are a very special area of study and although relatively little is known about them compared to language this is changing. They're very special, and over the last 10 years educational neuroscientists have begun to highlight some of these specifics of maths learning difficulties and this new understanding has resulted in part from insights about students maths difficulties especially dyscalculia and this more recent work has also just confirmed some of our older insights but we wouldn't want to say that this is just old wine in new bottles though, this is, in fact, new wine. Its new understanding that we are getting from an educational neuro scientific approach. So the first thing to say though is that maths curricula are remarkably similar worldwide. And this is a useful fact because it means that when we're studying maths learning difficulties we can look at evidence across the whole world and research that supports the understanding of the differences and we might think of the curriculum then as just a normative learning progression. As teachers we would expect the average child to traverse these steps in a similar fashion but what about those students who don't follow this normative learning progression? These are the students who are the subject of these presentations. So what is a maths learning difficulty? What do we mean by that? Well it refers to a condition that affects the ability to acquire arithmetical skills and it is often used as a broad cover all description for a whole range of differences that students have in their mathematics development. MLD students have difficulty understanding simple number concepts and have problems learning number facts and procedures. Dyscalculia though, like dyslexia, is thought to be a particular learning difficulty. That is to say just like in dyslexia you can be a poor reader and have dyslexia or not. Similarly with dyscalculia these students lack basic number sense and that impacts every aspect of their number processing. So even if student produces a correct answer or uses a correct method they might do so mechanically and without confidence and the severity of the mathematical impairment differs depending on the individual, however, the distinction between general maths learning difficulties and dyscalculia varies across Australian states and so as we go through these videos we're going to attempt to define dyscalculia for you in the context of assessment and intervention. Many of the defining features of dyscalculia can also be seen in students who do poorly at maths and the degree of the difficulties and the lack of impact of remedial intervention sets dyscalculia students apart from the other MLD students. So does maths matter? I expect you've been at parent teacher nights where a parent will come along and say “I was no good at maths, no wonder that my child is no good at maths" and “I’ve managed fine without maths”. Well there are some significant impacts for a lack of maths ability for students in our schools. So what sorts of things might they say about trying to cope with numeracy. Well they might say “In the mornings we do mental maths so she's saying like saying the questions and she goes really fast and then I then I stopped doing it and I'm left behind”. “I was paying attention to the question and then I didn't get it. And then the person next to me gets it and then I don't even ask him. And then I just like get confused and I get most of my answers wrong”. Now these are real quotes from real students who we have seen in our testing clinic. “I feel like screaming and saying why are you doing this? Why are you doing this? And I feel like punching the teacher”. They don't, of course. ”When I don't know something I wish that I was like a clever person I blame it on myself”. So all of these are the emotional or psychological impacts its not so much about being able to process maths its the impact on the students of not having a substantial ability at maths, and these have serious consequences for student welfare. “She just comes up to us and says ‘Ha ha you don't know anything. You are so dumb’ and then she asked me like questions like a thousand times a thousand which she knows I don't know. And it's very hard for us”. And of course other students notice. “Yeah, and then he goes and hides in the corner and nobody knows where he is and he's crying”. So what is the impact of MLD on students? Well in addition to not developing their skills at the same rate as their peers MLD students are at risk of low self-esteem, anxiety, depression, social emotional problems, behavioural problems, underachievement in areas of the school curriculum involving mathematics skills and limited school participation, to the point of exclusion. In my clinical private practice a high percentage of the teenagers who are presenting with these sorts of levels of psychological distress talk about difficulties in maths. It is a real issue for students. So many students report these emotional and psychological distresses as a result of their inability to perform in the maths classroom at the same level of their peers. And this psychological impact extends across multiple areas, it's not simply in the classroom because once they become anxious and worried about school it will have an impact across other subjects. Now the international prevalence rates for dyscalculia which is the severest form of mass learning difficulty are estimated at about 7 to 9 per cent which is roughly the same as the prevalence rates for dyslexia, although there's now some evidence that dyscalculia may be between 10 and 12 percent because we are now getting better methods of diagnosing it and also another difference between dyscalculia and dyslexia is that dyslexia is thought to vary between countries as a result of different language systems. A major recent UK government report concluded that developmental dyscalculia is currently the poor relation of dyslexia with a much lower public profile but the consequences of dyscalculia are at least as severe as those for dyslexia. I just want to point out why the description there is developmental dyscalculia. We talk about dyscalculia when we're talking about children developing into adults as having developmental dyscalculia. When we talk about adults we will often refer to it as a acalculia, but for the purposes of all these videos dyscalculia will mean developmental dyscalculia, and the purpose of these videos is of course to address this public profile of dyscalculia. It is a recent area of study. So this is why there hasn't been so much focus on it, but we are now becoming much more aware of the impact and the importance to get the information out. The report continued that until relatively recently dyscalculia has received significantly less a focus than dyslexia and this potentially sends a message that literacy is more important than numeracy. Again, that's what these videos are designed to change. So these five video presentations are going to unpack MLD with specific reference to dyscalculia from an educational neuroscience perspective and we're going to highlight what educational neuroscientists have discovered about MLD and the impact these advances in knowledge have in understanding maths skill development across schooling, and we're going to address the ways MLD and intervention strategies for MLD can be assessed and so that students can improve. So there'll be five presentations. This is the first of course and we're going to look in this one as an educational neuroscience perspective specifically focusing on core number abilities. These are thought to be crucial to understanding the specific maths learning difficulties students experience. Video 2 will extend video 1 and look at primary and secondary school maths. We won't have time to cover everything in the curriculum but we'll pick up specific areas that we believe have difficulties that can result in an impact on student progress. Video 3 is going to specifically focus on dyscalculia as a specific maths learning difficulty. Video 4 will look at assessing and diagnosing MLD and in video 5 we’ll look at intervention strategies and recommendations. Now I'm going to use the word diagnosis quite a bit and I'm not talking about clinical diagnosis here. What I'm talking about is understanding the specifics of the student's skills but we'll use the word diagnosis as that cover all description for just finding how how a student processes numbers so is number domain specific. What do we mean by that? It means can you have a number disability and not have other disabilities? So what has educational neuroscience told us about this so far? Well the first thing is that we and other creatures seem to be pre wired to compare two sets of objects and determine the greater number. So here we've got some canaries who can select a specific quantity regardless of the shape size and colour. Some salamanders can choose more up to a value of three. Pigeons can discriminate. Rats are very competent at number. They can estimate larger numbers and they compress a lever roughly 40 times to get a reward. Jack Daws can recognize the same number of different objects, parrots are able to learn the meaning of abstract numbers and symbols they can actually recognize some of the symbolic numbers. And of course the primates we know that primates have some significant number of abilities. Now if you were particularly interested in the number ability of fish there is a very entertaining and interesting YouTube clip there for you to access and it is Professor Butterworth, and it's a very interesting video but an important thing to take from all this evidence is that this supports the view that number ability is innate and also evolutionarily advantageous, and that does compare to dyslexia so dyscalculia and number ability are different from literacy abilities in this way. Children are born with number sense or core number ability and that allows them to match or discriminate numerosities visually but also in the auditory or tactile mode. So in hearing and in touching and we're going to start with looking at very early number development. So these are very young children. In fact they're newborns and they were familiarized with some sequences of sounds containing a fixed number of syllables and then they were tested with images which had the same or different numbers of items. And here it was four or twelve. So here's an example of the sort of familiarization that they heard 2 2 2 2 2 2 2 like that. And it was 4 or 12 so it could have been 2 2 2 2. So they were only looking at 4 or 12. And you might say well how can you possibly test newborn infants because they can't point to anything and they can't tell you? Well what we've noticed is with newborn infants if they're distracted or they're seeing something that is expected they don't pay attention to it but if something comes up that is unexpected or that they are surprised about they will look much longer at it. So the time they look at something gives us a measure of whether they think it's something they were expecting or something they weren't expecting, and these are what the trials looked like four or twelve and what they found was that the newborns look consistently longer at the displays that were incongruent. That means that they had heard four syllables and they were shown a picture of 12 but only when the numbers were separated in this ratio with three to one. So four versus twelve, or six versus eighteen. They weren't able to do smaller ratios like four to eight. But nevertheless it did show that newborn infants, and they were very young, were able to discriminate these numbers and compared to what they saw with what they heard. We also know that five months olds appeared to be sensitive to addition, only plus or minus one though. So the experiment that was done in this case was a puppet which was put on a platform and a screen was pulled up in front of it, and then the infant can see a person putting in another puppet. So the infant should now mentally be thinking there are now two puppets behind the screen. Sometimes the screen drops and there are two puppets and the infant shows no interest in that at all but if the screen drops and there's a single puppet there the infant looks amazed and looks much longer at this unexpected outcome. So it looks as those children as young as five months have this basic number ability. Recent research has suggested that our brains are predisposed to represent magnitude or number in two basic ways. An approximate way where we can discriminate numbers of objects so a pile of objects. Are they more or less than another pile of objects and also precise small number, and we're going to describe this as enumeration and I'm going to distinguish enumeration from counting here for a reason that will become obvious as we go on because one of the things that we're going to notice is that for some arrays you don't actually need to count. So we're going to talk about it as enumeration: saying what number is there. So these representations may reflect a single core system or perhaps two separate systems. But either way they are core number abilities and they're going to come up right through this series of videos. So we've shown that children and adults have different response time signatures for tasks that assess these core number abilities, distinguishing between expected and atypical development. So we can see when we are testing using just discrimination and enumeration we can pick up differences between children and adults in how long they take to process. And this is informative about whether they have a typical number development and although there are neuro biological underpinnings to core number ability we do recognize that there are also environmental influences in numerical development. So obviously we would need to discount some of those when we're doing testing of all students the differences in how we represent and process core number information effect are acquired or cultural number development. That's the number development that goes on in schools and that is actually the topic for the whole of video 2 it's a sort of further number development on which is built this core number ability. So this is what the discrimination task might look like. So this is a display of some blue squares on a yellow background, blue and yellow are known to be good discriminating colors, and the task is to say which side has the larger number of squares, and we've shown individual differences in student sensitivity to discriminate. numerosities in this sort of format in primary years and importantly we've shown that these differences are predictive of later measures of arithmetic at all ages. So this emphasizes the functional importance of core number abilities to the development of math skills and early diagnosis and intervention. The earlier we can find these differences and intervene the better the outcome for the student. So another another sort of core no ability is placing numbers accurately on a number line and this gives very important information about the student's ability to perceive the magnitude of symbolic numbers. So this though does require that the student already knows the symbols for numbers the numerals and so we don't always categorize this in the core number ability range although it is part of core number of course because it is looking at the spatial array of numbers and mapping symbols onto that accurately, and it's a quick educational tool to see if a learner has understood this relationship between a number symbol and a non symbolic form of the number which is a core number ability, and this is the sort of setup. You have a number line, zero at one end, one hundred on the other, and you put a mark where sixty four goes, and we've shown that young children tend to underestimate the position of the larger numbers and overestimate the smaller numbers. So this would suggest that they have an ordinal representation of numbers, they do understand that they go in order but what they don't understand is the spatial distribution of those numbers across the line and the children and students often do not progress to accuracy on this task and they in turn exhibit poor arithmetic abilities compared to their more able peers. So this also seems to be an important predictive test and the degree to which the students response deviates from the correct placement of the number is both diagnostic of their understanding of the relationship between numbers and also predictive of later success, and this is a study done by Bob Reeve and Brian Butterworth, with one of the postdocs, Jacob Paul, they had six year olds and tested three times over 18 months and they had in the first phase just a zero to 100 number line and some numbers that they had to place on that line. In time too they had a zero to one hundred and a zero to a thousand number line and in time three they just had zero to a thousand. So this would be an example of a zero to a thousand number line and what the the group were interested in was the magnitude of the errors or how much the position that these student marked on the line deviates from the correct position on the line and how this declined over time and what they noticed was that students definitely become more accurate over time. The error times though on the first exposure to each of the different line lengths were related. So what we had was a type of error pattern on zero to one hundred at time one and that got repeated on the zero to a thousand at time two even though there had been improvement on the zero to one hundred line time two, and the individual differences in number line error patterns predicted performance on a mental calculation task. So this looks like it is an informative test for understanding individual differences in this particular type of number ability. This relationship between the symbols and the space that numbers can occupy. So enumeration then. So enumeration simply involves asking the learner to name the number of objects in a display, and this is measured in terms of accuracy and speed. So the sorts of stimuli we might have to measure, the exact number system, might look like this. This is why I'm referring to it as enumeration rather than counting because that leftmost one of three, almost no one except for someone with dyscalculia will count that, they will just say three. So this is an important feature of numerosity and enumeration, that the time to enumerate smaller arrays, those less than or equal to four, compared to that time taken to enumerate five or more dots, and what we find is that students and adults typically enumerate small dots, small arrays of dots rapidly and accurately and are slower and more error prone in enumerating the larger arrays where they are counting individual dots and this apparent automaticity is called subitising and subitsing turns out to be a really important indicator of core number ability. So how is this significant then. Well, first of all our extensive work examining students’ enumeration has highlighted a high degree of variability in response time and just within a single age cohort, and it's a very simple task. You might not expect to have such variability but we find we have significant variability in this task. Moreover the individual differences in enumeration performance reveal distinct profiles of subitising ability, that are stable across time even though the students within the groups get faster overall, the differences are maintained and notice I'm using the word profiles here. We use a different sort of analysis, we're not talking about cutoffs here where we think of maths learning difficulties as an all-or-nothing thing. We're talking about profiles, we're talking about differences between different profiles and specifically in each of our studies a group of students have been identified who have a very short subitising span: up to three dots, and in fact some have no subitising at all, they count even when presented with only two dots, and the students with a short subitising span also exhibit poor performance on other tests of maths ability, and this emphasizes how important this test is for understanding a student's core number abilities because it has such an impact on their future performance, and we're going to cover the contribution made by enumeration abilities to identifying MLD and dyscalculia in much more detail in video 3. So what develops then, once we have these core number abilities? What develops after that? Well, we've shown that children are born with core number abilities that are variable between individuals but stable within an individual, but what is the impact of the individual differences? How do they affect future maths performance? So from the building blocks for numerical capacities, the core number abilities, estimation enumeration, children develop learned or cultural number abilities, they learn them in schools and they might look like this. So we're going to distinguish between a capacity and an indicator, capacity is what a student “could do” in ideal conditions. An indicator is what they “actually” do. So numerosity is a property of sets, the fact that students can understand what the numbers mean as in a cardinal fashion, so if they count to five they understand that means they set all five objects. Another capacity is estimation which I've talked about already, being able to estimate number line is an example of an estimation task, but also if you are just comparing piles of counters, that's an estimation task as well, and then a sense of ordering. We need to know that the numbers are in a particular order, one is larger than the other, and then we have these acquired cultural tools which we learn in school, and those are the things that need these core number of abilities in order to make progress. So the indicators of these particular capacities are enumeration, magnitude estimation, number comparison, number line mapping, and things like counting and calculation, and so when we get to the assessment video, which is video 4, we will be looking at these indicators and what they indicate and how you can interpret them. So what develops then? Well, the milestones in early development of arithmetic competence look something like this. Now at zero, we've already we've already shown that newborns can discriminate on the basis of small numerosities, four twelve, we saw that in an earlier presentation. They can add and subtract one by four months. They can start to discriminate between increasing and decreasing sequences by 11 months. At two, they begin to learn the sequence of counting words and can do one to one correspondence in a sharing task if they're asked to share toys between two of their friends for example. Two and a half they start to recognize that number words mean more than one so they don't just if you say can you give me five of those they don't just grab five of them. They know that they actually mean something specific. At three, they can start to count out small numbers of objects. At three and a half, they can add and subtract one with objects and number words, and some of them will have this cardinal principle, this idea that the number when you count you're also including all the others in the set, and at four they can start to use their fingers to help them with addition. Now this is a typical development, this does not mean everybody can do all of these at the same time but there are a high proportion of children who can, and the reason for raising this is how much a student accesses number prior to starting school much more than in reading and writing. They are accessing and working with numbers long before they get to school, so young children then arrive at school with a starter kit, and this contains tools that they can they can use to bring to the task of learning arithmetic, and counting aloud has the potential to make a bridge between this innate capacity for numereosity to more advanced mathematical achievement. Learning to count takes about four years to fully master and errors occur along the way, but it's important to note that these areas are randomly distributed across the population and they aren't significant or predictive of later maths difficulties. So examples of errors might be these double counting where a child will go one two on one object and then three or four on the next, omission where they go three and then they count four but they don't touch an object and then they get to five, or split counting where they have a single word split across two objects. So children start to count around two years old and then they progress in stages until about six years old when by then usually understand how to count and how to use counting in a near adult manner. Two year olds though showed a range of counting tags and then it's really important to keep this in mind when we're talking about typical development. This is the variety of development that we see in a range of two year olds. So you can see that in when they're counting two the number of them said one some say five. Note though that five a lower very variability, sort of back down into the 2 and 3 range, compared to the others and the reason for this is if you ever ask a child who is less than 5: how old are they? They always answer by putting up fingers, and interestingly they'll often put up fingers and not use the word. They'll just tell you with their fingers how old they are. So they're very aware of their hands by the time they get to three or four, and that's probably why there's less variability at five because of their five fingers, but the variability is important because when I'm talking about typical development I'm not ignoring the fact that there is a wide range of variability. Counting then, well cardinality principles, they also vary as a function. So this is this idea that if you count up to a certain number it includes all those previously and you can see here that three year olds are reasonably accurate at two and just diminish after that. Five year olds are reasonably accurate up until five, up above the number five and then they diminish. So these skills do take time to acquire. Now what about preschoolers? We've already talked about preschool a little bit and we find that these preschool years provide a foundation for school maths achievement. Counting skills are a child's earliest contact with symbolic number, and with this understanding children are able to process precise relations between number symbols, such as ordering and comparing symbolic quantities. Many three year olds are capable of processing relationships between small symbolic numbers like up to four and a number of five year olds will be able to process relations between large numbers, between five and nine, and numeral knowledge begins to develop in preschool shortly after children learn to count. I mean by that the symbolic numeral and the numerical meaning of count words. By the age of four approximately a quarter of children can identify the numerals 1 to 9. Although some children as young as 18 months can identify the numerals 1 and 2. Now preschoolers show some success on number line and number comparison task particularly in the 1 to 10 range, and research has shown that children as young as two succeed at computing non symbolic addition and subtraction with blocks and counters for example but only in small sets so they have to be sets less than three items. Preschoolers are much less successful at solving purely verbal problems, that is to say without any without any manipulatives, but it does increase across preschool and it is also important to note that these abilities present in preschoolers prior to instruction, exactly mirror those competencies that we see developing later. Early maths competence though is not unitary it composes and it comprises a number of component skills, but is there variability in preschoolers maths performance, and importantly is it indicative of later number processing difficulties? Now this is a study done by Sarah Gray one of our team and Bob Reeve in line with core number abilities that we described already and the impact of variability. Graham Reeve looked at 103, 40 to 60 month olds across six math measures. Now they were: the ability to count count sequence how high up one of these children can count, object counting enumeration, give a number so you ask the student can you please give me five counters, naming numbers, it's important that they are able to recognize the symbol and be able to name it, ordinal relation so comparing the size putting numbers in order, and arithmetic in here just non symbolic additions. So using counters to do some fairly simple addition and again we didn't use a cutoff or an averaging analysis for this, we use this partitioning. So we can look at particular profiles of performance, and in this case there were five different profiles and we could see here that there were different profiles of strengths and weaknesses as well, and this is an important part of this approach to analyzing the data because it means that individual students don't all just have the same pattern in decreasing order. It means that we've got some specifics within a particular group. So the five groups were called Group A: excellent en masse ability on all tasks, B was group good en masse ability with relatively good arithmetic ability, C was good en masse ability but relatively poor count sequence, D was average on all and E was poor ability on all. Now I'm going to spend a little bit of time on this rather complex looking graph here. First of all the y axis, the vertical axis is from zero to 1. That's just the proportion correct. So 1 means one hundred percent correct, and 0.2 is 20 percent and so on, and these at the bottom are the six tasks that they were given: count sequence, object count, give a number, naming numbers, ordinal relations, and arithmetic, and the five groups are colored accordingly on the legend, and you can see that the excellent maths group did well on count sequence and quite well on object count but not as well as they did on count sequence or give a number or naming numbers. You can also see that good at maths for count sequence, good at maths did very well on give a number, and quite well or reasonably well on naming numbers but relatively poorly, interestingly, on count sequence. So this highlights how these different patterns of strengths and weaknesses can be really useful for us when we're looking at interventions. So three of the maths profiles showed similar ability, overall excellent, average, and poor, but the remaining profiles showed this unique pattern of strengths and weaknesses and those patterns are going to be informative for how we intervene to assist those students and this good maths poor can't sequence profile suggests it's possible for children to show strong ability on more complex maths tasks and yet perform seemingly poorly on more basic skills and this is inconsistent with the claim that no particular maths skill is a necessary prerequisite for another skill and findings also show that different cognitive markers are associated with different preschool maths, but despite that dot enumeration and something called spontaneously focusing on number, what that means is if you show a child a picture and you say “Tell me about the picture" or “Describe the picture” if they focus on, well there are three ducks in the picture or they say something about numbers in the picture we would suggest that they have spontaneously focusing and those were found to be stronger markers of good and poor maths profiles and the research showed that both dot enumeration efficiency and response time predicted non-verbal arithmetic ability over and above working memory and something called response inhibition which which is a cognitive measure, and so that means that these numerical measures are more predictive of future maths than the cognitive measures and the fact that the RT profile characterized by a smaller subitising range in profile C was associated with poor addition performance is important because this is exactly the same finding we find in school age student which we'll describe later. It suggests that a weak subitising profile has potential as a diagnostic marker of emerging maths difficulties in preschoolers and so if different number competence profiles can be identified in preschoolers, how do these sorts of basic number of differences play out in primary and secondary school students? That's going to be the topic for video 2, but before I conclude I'd just like to reiterate some of the messages from video 1. The first was that less is known about number difficulties than language difficulties. We're focusing on an educational neuroscience perspective and the insights that we can gain to give scaffolding for students with difficulties in number. We've of course emphasized the variability in maths learning difficulties and also that core number abilities are functionally relevant in three ways: they're relevant diagnostically to find out what what a child or student is able to do, for developing assessment tasks that we make sure that we are measuring that ability accurately, and for informing intervention strategies.