Learning Difficulties in Numeracy – Intervention Strategies (5/5)

Welcome to the final in a series of five videos that have been developed in the collaboration between the University of Melbourne and the Victorian Department of Education and Training These videos have been about maths learning difficulties and particularly dyscalculia This final video is going to build on all the knowledge that we have acquired along this journey together and culminating now in intervention So in this final video we’re going to draw on all the ideas we’ve used in the earlier presentations and provide some intervention advice for students with maths learning difficulties More specifically we’re going to draw on our own clinical intervention framework to offer suggestions In large measure our framework was based on interactive assessment This is a model of learning which aligns with the so-called “Vygotskian approach” to instruction and development Now Lev Vygotsky was a psychologist in the early part of the 20th century Vygotsky believed that children were capable of much more advanced thinking of problem solving when they’re assisted by someone than when they’re on their own So that is to say a more experienced “other” provides careful help and support for the learner An interactive assessment is used both diagnostically as an assessment and also as an intervention measure, and I reiterate here when I’m talking about diagnosis, I’m not talking about clinical diagnosis, particularly. I’m talking about understanding the strengths and weaknesses of individuals so that we assess those and intervene appropriately Now the term interactive or dynamic assessment is used to distinguish from static assessment measures which are normal school based or other area tests. Well, what happens when you have a static assessment is your assessing the performance of a student at a particular point What you’re not assessing is how they reached that point and anything about the learning along the way and in particular if the student got the problem wrong, what they didn’t understand Where did they go wrong? What were the things that caused them not be able to solve the problem? So it’s often claimed then that the amount of help and the type of help that a student needs from interactive assessment is a useful way to identify different types of learning difficulties, and this is particularly useful in our present context because it’s going to distinguish between different types of MLD So what is dynamic or interactive assessment? Well, it’s simple It involves developing a mini learning progression within any kind of relatively narrow maths domain like simple word problems So just the steps that would be required for an experienced person to solve the problem are first determined and then as the student progresses through the problem then the experienced other person will provide guidance if necessary, provide scaffolding if the student gets stuck and so on So the steps required to solve the problem have to be first specified and the focus is on the interaction because the teacher and the student collaborate to solve the problem Now the student might not have the competence initially to solve the problem on their own but with the support for completing steps across problems from the teacher saying things like: “What do we do now? What did you know from the problem? What is the problem asking you to do?” the responsibility to solve problems is handed over to the student So the nature and the amount of help the student requires to solve problems independently, we would say was the assessment side of this, and the ability to transfer across problems, which is a known difficulty for dyscalculics, provides an index of what is learned and a measure of learning potential The long term goal of interactive assessment though, of course, is for the student to internalize the problem solving concepts and procedures in order to solve problems independently but an important point about interactive assessment, is all students complete all problems because they are helped along the way However many times they stop and need help, they finally solve the problem, and this addresses some of the psychological issues that students have with problem solving because they also see So the principles then that are guiding our model of intervention support are that performance

variability and individual differences in development trajectories necessitate different intervention approaches: one size doesn’t fit all when we’re looking at intervention and support for learning We need to be able to accurately assess a student’s core number ability and we’ve talked about assessment at length in video four now for their enumeration and their estimation including number line estimation, which we talked about as a classroom based activity, their arithmetic abilities, from counting to addition, subtraction, multiplication, division, their fraction abilities, particularly assessing the errors they make and their algebra abilities and in algebra we are interested in their understanding of algebraic reasoning Now as I said earlier there are many competencies along the maths developmental pathway I’ve focused on these core number and arithmetic for primary school are imperative for future maths success, and then two well-known areas of difficulty, fractions and algebra, where these difficulties may be highlighted again, and again diagnosis via assessment is critical to determine the appropriate intervention So students with different kinds of difficulties, they require different kinds of learning support and the optimum intervention for an individual is based on their specific skills determined from assessments and these outcomes will help also to identify the difference between deficit delay and expected maths learning pathways and just referring back to the video clip in video three from the businessman who had a lifetime experience with dyscalculia, this isn’t going to provide a cure but we can have coping strategies and the hope for these intervention strategies is that this is the building of some coping strategies for students with dyscalculia So Fuchs and colleagues in 2008, articulated these seven guiding principles for intervention which are are important for us to take into consideration as we work through these interventions First of all interventions must be instructionally explicit Students need to know exactly what it is we’re asking them to do and then we need to minimize the learning challenge Now interactive assessment does this very well because it breaks it down into the smaller steps but the student needs clear and precise, logically ordered explanations Breaking it down into smaller parts always will help the student because it minimizes the amount that they have to learn, and also it will isolate their difficulties It’s imperative too that we have a strong conceptual basis for any procedures, in previous videos I have talked about that students can apply a procedure, they can learn a procedure but if they don’t have a conceptual basis they are going to forget that procedure So this conceptual basis is important, if you’re introducing a procedure, making sure that you have presented the conceptual basis for it and reiterating that continually as the student uses procedures The student will need to practice Skill practice is essential, especially if the conceptual understanding is going to result from repetition of a procedure We need to need to review where we’re at now, where we’ve come from and where we’re going to go to, we’re going to need some motivators to help students regulate attention or behavior because often students by the time we get into these intervention strategies we find that they have been turned off maths altogether Partly they’re anxious about it Partly they just find it too difficult So it’s important that we motivate them and make it fun for them so that they do succeed, and that’s that’s an aspect of the “success for everyone” type of approach We also need to monitor progress continuously We need to know that the student is progressing and if they aren’t we need to go back to see what it is that we need to intervene over So videos 1 and 2 then outline the development of children’s maths ability from infancy to pre-school to primary and secondary formal ed instruction Difficulties for MLD children,

students were described in each of the areas covered and video four described many different measures of the ability reflecting the different stages of maths development that we described in videos one and two. In this video, we’re going to consider intervention in four areas, in turn: core number, counting and arithmetic, fractions and algebra Now I’m going to not take a long time over this but just to remind you of the sorts of difficulties that we’ve raised I’ve talked about a short subitising span, I’ve talked about difficulties with mapping on the number line, we’ve talked about counting difficulties, difficulties in multi digit addition, difficulties with applying arithmetic knowledge in fractions and algebra All of these are going to be addressed during this intervention so that we can see how students can be helped to overcome the difficulties and before I go any further, I just want to dispose of this hoary old chestnut and that is: rote learning and conceptual learning We’re not going to distinguish between the two I’m not going to talk about rote learning or conceptual learning for the following reasons The support and the help that we give to students and their expectations is going to depend on their abilities Sometimes a MLD student or one with dyscalculia will need to be shown explicit well-described maths procedures several times before they begin to internalize that procedure and they may need lots and lots of practice of that skill So long as you are continually justifying why that procedure is being used in terms of the conceptual understanding then eventually internalizing it will happen and this process of internalization could be thought of the learner appropriating explicit learning behaviours for their own purpose So they come to see the importance of understanding what they’re doing as they perform particular procedures, and this in turn may be thought of as the first steps in constructing a conceptual framework So rote learning may be the first step particularly in dyscalculia where we are needing to find other ways to teach, dyscalculics have difficulty because of this mapping difficulty, that isn’t going to go away So we have to find other ways to teach them and I’m going to also talk about an intervention strategy that actually specifically addresses this mapping to build that knowledge in dyscalculics So there are many maths interventions, support resources and sources of advice A number of organisations provide paid support to putatively help children learn maths so there are online curricular software, board games, maths books and programs specially designed maths component software and the like Indeed, between the 1870s and 1970s over one hundred and seventy books with the title “Why children cannot learn maths?” came out… were published How do we navigate this overwhelming array of support learning intervention options? Well, without the support of a significant other, students are often repeatedly asked to solve problems that they cannot solve We find that students with dyscalculia will report that they have been asked to do many worksheets and they don’t know how to start, and then this leads of course to failure, frustration, anxiety and sometimes a fear of maths So we need to link all that we’ve talked about in these videos, the assessment diagnosis and intervention, so that we can get the best outcomes for all of these students Now we live in a digital universe and students as you know are far more digitally aware than those of us that are older, and this online support is useful for identifying the range of mathematical phenomena and we need to encourage students to use these, as it’s a familiar medium They are used to working online and online programs often include dynamic displays and students actively interact with the program, so that the program is building on the student’s knowledge, but is this enough? The importance of learning maths depends

on extracting principles and linking dynamic concrete representations with abstract maths ideas and that may not be, if the student is going to just work on the online program on their own and unassisted. However, if we have external support from a more experienced person to ensure that this understanding is being realized and that the student is understanding what they’re doing as they’re working through these dynamic arrays the outcomes may well be different and that support person is going to focus on generalizing across problems and transferring to new problems Two important steps that need to be made when students are learning how to work with certain problems they need to be able to generalize it to other problems and transfer to new problems Students also need to be self-motivated and have increasingly less support, but that isn’t going to happen until they are able to is what is being presented to them and how they are to interact with it and the other thing to not lose sight of is that maths is implicitly embedded in everyday activity One very useful intervention strategy is: you’re driving along the road, the speed limit changes and you say to your dyscalculic son or daughter, am I to go more slowly or faster? Simple things like every day It’s a little bit like hiding vegetables in hamburgers or beef burgers Students won’t realize that they’re actually processing with numbers if you’re introducing them into their everyday life So let’s now look at intervention in the core number area Students enter school with a considerable amount of number knowledge The home learning environment in early years can promote or inhibit the development of arithmetic competence though and that may be why we have such a variety in preschool students when they’re entering school The frequency of number activities in the home has been associated with maths skills at the start of school and then into grades four and eight, and this is part of this Trends in International Maths and Science Study data from 2012 So there is an important contribution that the home is making to outcomes in schools Numerical activities have specific effects over and above parent education and socio economic status According to the same study and children who start school with poor knowledge and skills in numeracy are unlikely to catch up to their peers Now that’s a quote this paper along with individual differences in numeracy skills are evident at school entry prior to formal instruction, suggesting that children acquire fundamental skills at home Now what this is saying is the evidence is that students who are entering the school are going to be somewhat disadvantaged and the whole purpose of this intervention video is about that: if a student when they enter school is already having difficulty in numeracy then early intervention is the answer, but equally should we be assisting parents in optimizing children’s number skills as early as possible? If there is some disadvantage from what they learn prior to school perhaps we need to do some work in making sure that all children are entering school with a certain basic toolkit So before we start to detail specific intervention strategies, let’s just try this So I’d like you to please solve the following arithmetic problems Here’s the first one There’s the second one and there’s the third one Now I won’t make you go on with that because I suspect that for many of you, you didn’t know where to start with it So what might have helped? Well, if you’d known that that was the symbol for five and that’s the symbol for 11 you would have been able to do that sum easily and you would get the answer sixteen but you then need to know what the symbol is for that If you’d known that this is twenty four and that’s seventeen then you would know that that is going to be seven, that’s subtraction and you would again need to know the symbol, and if you knew that that was two and that’s nine you’d be able to know that that was 11 and you’d insert the symbol that you can see that for 11 Better still though if you knew that those symbols represent those arrays and you could relate it,

then that would make life much easier So what you’ve just experienced is what it is like for a student with dyscalculia in a maths classroom That is how they feel when symbols are presented on the board and they’re asked to do things with them but they don’t know what the symbol represents So we’ve described dyscalculia as this deficit in the most basic capacity for number, this number sense, and on which everything is built in maths For dyscalculics the relation between number and symbols, words and arrays is not secure, without number sense a dyscalculic struggles with the principle of commutativity, that is: three plus seven equals seven plus three or with the relationship between addition and subtraction if five plus three is eight then eight minus five is three and eight minus three is five We’ve already emphasized the importance of an accurate assessment of students specific maths ability Students are different and the support they need will also be different Prior to any intervention, assessments designed to identify MLD students in terms of delay or deficit are paramount and an understanding of number concepts, number structures and calculation procedures help support discount heretics in the difficult process of trying to learn and remember them So this means then that any intervention for maths difficulties from core numbers to understanding right through to the higher maths abilities has to begin with activities designed to strengthen number sense For all students going back to the basics is going to be an important part of making progress later So one number difficulty then is this abstract nature of number We talked earlier about the arithmetic difficulties experienced by students once problems ceased to include pictures and moved to that So what helps cement this relationship between number, symbols, words and non symbolic array? In the early stages guide pupils using concrete materials for example and then use them to split and combine sets of objects Students move on to solve problems without manipulatives, but it’s important not to move on until you’re secure in knowing that they understand what they are doing with the manipulatives but keep them still available to check the answer or assist when stuck, and then of course digital activities can also be used, but again they need to be adaptive They need to use the learner’s previous performance to determine the next task So several interventions have been developed that target these early numeracy skills including something called number sense interventions by Jordan and Tyson These are evidence based interventions supporting the development of key math skills such as oral counting, number recognition, numeral writing and there are 24 scripted lessons in there about half an hour each and the specific skills that are being developed here are recognizing quantities and numerals, making associations between numerals and quantities, writing numerals, solving story problems and solving written equations, but these are these are the sorts of tasks that are going on in school all the time and so you don’t necessarily need to go and get a specific version of this It’s available by just looking through the worksheets that you work through and it might include something like this: where you have four objects and you have to select whether it’s for two or three or the student has to select… all of these sorts of things quite easily replicable in schools Board games such as snakes and ladders are incredibly useful In the video clips in video two on dyscalculia, you might have noticed there was a snakes and ladders board on the table there They are used often in intervention for dyscalculia and the reason is twofold One is the dice that they have to shake to move have non symbolic arrays, but the game depends on counting the relevant number of squares expressed on the dice which means the student needs to be able to relate the non symbolic array to the numeral and then count appropriately and then the other thing about snakes and ladders board is a little bit like a number line it’s presented in a sort of number line fashion and it shows the relative positions of numbers along that line in an equal spacing so magnitude representation, spatial ability, number line abilities are all being reinforced through snakes and ladders

Specially designed online programs to increase understanding of the mental number line have also been developed and it might look like this: the grasshopper jumped forward seven and backwards six choose the matching sample, again this can be replicated easily in the classroom and the aim of digital interventions, as with teacher interventions, is to make numbers meaningful for the learner Recently Professor Butterworth and his wife Laura logged in to develop this game called number beads and it uses this level progression approach to try and build this mapping between the known symbolic form of a number and the symbolic form, and if a child is thought to have dyscalculia, that is, they have this poor number sense, it’s likely that they will need help in this decomposition and composition of sets or groups and also with approximation So the idea here is to use a method of vanishing cues So it’s a little bit like interactive assessment but instead of having an adult supporting, these are cues that are presented to everyone at the start and they diminish as time goes on So the first presentation then is a coloured set of beads, no digits to be seen, and in fact the colored strings are… each color represents a single numeracy, so the target for here is four, and the idea is that the student has to make groups of four out of what they’re presented with So they might divide that into four and it will separate and make two sets of purple four or they might cut one off that array and you will then get one white and a four purple They now find that they have a one and a three that they can combine and that will make another four, and as they run out of beads, more beads come onto the screen for them to successfully make up to ten of these combinations Now the next set of screens once they’ve completed those ones, the target is still there and it’s still coloured but it now has a numeral with it so that the student will start to associate the numeral with the target and the task is exactly the same You’ve got a target of six so you’ve cut two off the eight and one off the seven and so on and then you can combine ones that you’ve got left over into sixes Then it goes into white beads, in case as that cue of colour goes away so the student is not now looking at the colour, they’re looking… in fact their attention is now much more drawn to the symbol than the array because the colour has gone away. There is a stage four, where there are digits but when they split they do become the beads but the last one is simply digits So through these five stages and there are quite a few repeats within the stages, quite a few levels, the student has moved from a totally non symbolic virtual vision of the number into a totally non symbolic into a totally symbolic with some cues that help them to make this transition and there is clearly success The measurements that are coming out of this would suggest that dyscalculic students are forming associations… the brain is forming new associations to join the non symbolic form with the symbolic form, which is this important mapping that all students need to make progress Dyscalculics are able to make better sense of number when the teachers use concrete materials such as beads, counters, and toys, composing and decomposing sets of blocks accompanied by counting helps dyscalculic learners understand the foundational concept that numbers are made up of other numbers and using concrete objects provides a presentation of the relationship between addition and subtraction and between multiplication and division For example, if we take 30 counters and share them between five children and then we take the same counters and share them between six children, what does the student notice? Now they might not notice anything the first time you do this but if you do it often enough they may come to notice that five lots of six and six lots of five come to thirty So the aim is to draw the student’s attention to these patterns of relationships between multiplication and division and how a single number effect can be used to develop other number effects So, for example, students with MLD may need many examples of this sort before

they’ll notice and internalize that five lots of six are thirty and six lots of five are thirty and then they might find in the division domain that if you take 30 objects and you divide them between five people each will have six and you divide them between six people each will have five, they will need many repetitions of this but one would hope that eventually they will start to instantiate some of this knowledge Similar activities can be used for addition and subtraction but the important point to raise here is that dyscalculics often seem to understand a concept one day and forget it the next and the likely reason is that they’re learning procedures without connecting to any kind of conceptual basis and it’s the conceptual basis, and that this issue of checking that they understand and they fail to remember the list when they presented the next time if they haven’t got a conceptual understanding So this is why we need to continually check that they have this understanding and this brings us back to dynamic assessment with a more experienced person, the intervention doesn’t rely on the student making assumptions about relationships, they are being pointed out to them and often we rely on a student to make a connection between the process and the deeper level of understanding where, in fact, that’s not a step they’re able to take unassisted So this support person can encourage the student to talk through their thinking processes Think back to Trish She talked about the importance of talking through as you are working through and then you get this idea of how they’re understanding the learning process This provides detail about their specific learning difficulty, and it also supports in mastery of the task which is most important Now let’s consider this weird problem Josh’s mother buys him five pairs of socks, one pair cost two dollars eighty five but a five pair pack costs six dollars fifty How much does she save by buying the pack? Now an interactive assessment is particularly valuable in this kind of problem because the problem can be broken down into the steps that are needed in order to solve it and the student might not know where to start So the support person would ask questions like “What is the question asking you to do? What do you already know?” and successful word problem solving involves these several processing steps and the difficulties experienced provide information about how the student has understood the problem Now students might need help to overcome a number of difficulties along the way within a single problem and this is what will be provided through these specific prompts by the support person Now I will be talking about interactive assessment in more detail, the specific problems towards the end of the video, where we talk about algebra word problems Multi digit subtraction intervention again, this was an error that we pointed out and note again, I’m going to talk about concrete materials The use of concrete materials is really really important for students with maths learning difficulties So here’s the problem: three hundred minus one hundred and thirty nine The first thing the student needs to do is to get out, three one hundred blocks and display it and then they’ll write the problem in a column aligned format Now they notice immediately that they can’t take 9 and they can’t take 30 So the first thing they need to do is to trade one of their blocks for ten 10 blocks, and they need to then note that in their question here They’ve taken one of the threes, that becomes a two and they’ve put ten in the ten 100s, so they’ve got two 100 blocks and they’ve got ten 10s The student then needs to trade one 10 block because they still can’t take away that nine So they now trade 1 10 blocks for 10 unit blocks and they again note the trade on the question So now they’re in a position to be able to do all the subtractions, they can take nine from ten, they can take three from nine, and one from two, and that’s precisely what they do, and they then can see what is left, which they would then write in the solution So using these blocks to do that kind of subtraction addresses this error of not remembering… if you just do it by rote and you cross out a number and you move things along, you’re not going to necessarily do this error free, if you use the blocks you are very likely to get the question right, and also to have a visual picture of what’s happening So in summary then, intervention is based on understanding basic number concepts, core building blocks,

the sequence of activities is carefully structured to adapt to the learner’s current level of understanding, instructions are explicit with a strong conceptual basis The learning challenges should be minimized, so in clear, precise, logically ordered small steps, frequently checking that the student understands and revisiting the same concepts is imperative, the intervention should make arithmetic learning a positive experience, if they can achieve an answer for every question, they are going to feel more positive about it Learning should be fun, motivators to help students regulate their tension and behaviour, and now we move to fractions, which we talked about as one of the areas beyond primary, although they are introduced in primary schools, where problems occur So they are important in and beyond school and many students don’t understand them What’s the best intervention? Well as with all other interventions I’ve discussed using concrete materials so that they can can connect the physical forms of fractions to the written fractions is as important as connecting symbolic numbers to non symbolic arrays It’s the same kind of thing but you need to be aware that if you only use circles or you only use fraction bars, students won’t be generalizing their learning, so you need to do it in a variety of concepts When students count fractional parts they need to do this using manipulatives, on number lines, or orally, and this will help them understand fractions are actually numbers So explaining the meaning of fractions to students using clear language, for example, explaining that three over four means three of one quarter units Which means you’ve got one whole you’ve divided into four Each one is a quarter and you take three of those quarters Or it could mean one quarter of three wholes and a useful explanation for understanding fractions is this relationship between the division sign and a fraction, and that is that a division sign is a fraction with no numbers added, and if you think of it that way then you are going to get this rational number approach to fractions, that two thirds means two divided by three, because it’s like a division sign, only you’ve put some numbers into it now So you can distinguish where we’ve got one third of each of two holes or two thirds of a single whole, and those two things mean exactly the same thing and those two diagrams mean exactly the same thing, and it helps for students to understand the way in which fractions are actually constructed from the numbers It also helps a greater understanding of what happens if we count in fractions, and so for example you can count and get the same numerator and denominator to get improper fractions when you go beyond the whole So for example we could start with two fifths and count up one more fifth and then we got three fifths Now this is something that is often missing in students understanding, that they don’t understand that two fifths and one more fifth is three fifths, and then they can count on, four fifths, five fifths, which is one whole, and then six fifths which is beyond the whole So that students are going beyond the single unit which is often used in fractions and they understand the pattern and this will form the foundation for adding and subtracting fractions An inherent source of difficulty is understanding fraction magnitudes and understanding them can make a huge difference So encouraging students, and remember back to an earlier video where we talked about this difficulty of one third plus a half equals two fifths Why can’t that be correct? You say to a student and then by getting them to cut up objects into thirds and halves and sticking them together they can see that it’s much more than a half already, so two fifths can’t be right because that isn’t even the half, and then you might encourage them with some plausible answers, you might say “Well, supposing we were to cut up this circle into sixth and then we’ve got a third, which would be two sixths, and 1/2 which is three sixths and that’s five sixths That’s nearly the whole So it can’t be two fifths Lastly we’re going to just look at some algebra intervention and in video 2 we described the types of difficulties students make when they’re making this transition from arithmetic to algebra,

and we highlighted four areas: use of variables, replacing numbers, where arithmetic has a specific numerical answer algebra has a range of answers, difficulties with the equals sign, our arithmetic interpretation as a uni directional left or right operator replacing gives or yields or results in, solving algebraic equations, that arithmetic processes are based on this uni directional left or right interpretation are inappropriate for equations when unknowns both are on sides of the equals sign and solving algebraic word problems These are more complex than arithmetic word problems because they involve constructing an equation to represent the mathematical relationships first Now many students have difficulty with the transition from arithmetic to algebra, but the problems are magnified for students with MLD who don’t have a well grounded understanding of number and arithmetic So algebra often involves manipulating expressions based on a sound knowledge of the structure of the expression built through arithmetic experience So just to reiterate, students with maths learning difficulty and dyscalculia may need to step further back before embarking on these algebraic interventions So after many years of experience with arithmetic problem solving, students often interpret the equals sign in this uni directional left or right way and early algebra is very similar to arithmetic and arithmetical procedures can be correctly applied in the algebra and that really is what leads to these difficulties For example, this equation P plus six equals eleven, is basically an arithmetic equation of the sort primary students have seen many times with a box in place of the letter Problems like this reinforce the arithmetic, one directional interpretation of the equals sign which later presents problems for algebraic problem solving. However, the same problem could be presented like this: eleven equals P plus six, it’s exactly the same problem but presented differently and this would assist students with the transition to a bi directional interpretation of the equal sign for algebra Now students when they first see these will express concern They will say it’s back to front I can’t do it but this is an opportunity for teachers to explain the meaning of the equal sign and better still when arithmetic is being taught Also presenting the equations that you present with boxes and arithmetic, presenting those in the other way round will also challenge this idea of a one directional operator for the equal side, and it will make a large difference to students as they’re approaching algebra So interactive assessment is particularly valuable in algebra particularly in word problem solving and where there may be many areas that the student has difficulty with and we need to know what those all are So diagnosing these specific areas of difficulty when you’ve got four possible ways the student can go wrong in algebra is very important There’s not going to be time in this video to address interventions in all four areas but I’m going to talk about algebraic word problem solving because this actually encompasses the other areas that present difficulty and it involves three problem solving processes really: comprehension, where you understand what the mathematical relationships are in the problem, what is the text of the story problem telling you, a representation phase, where you represent the information that you’re given in some form of symbolic language, usually an equation, and then a solution phase, where the equation constructed in the representation phase is solved to generate a numerical solution So let’s consider this problem: Julie’s 16 year old brother is 8 years older than her How old is Julie? So we need to convert this into a symbolic form and what we’re going to do is… we can’t just look at it and do a word order… make an equation out of it and this is a particular challenge for MLD students because they need to understand the maths underlying maths relationships To construct the equation, the unknown part of the problem is first expressed as a variable, then the remaining relationships are built So if Julie is X years old, how old is her brother? He is X plus eight because he’s eight

years older than her We know though that Julie’s brother is 16 years old So x plus eight equals sixteen, and so this simple arithmetic equation gives Judy’s side age as 8. So some students might find all these steps difficult to negotiate and interactive assessment provides a series of hints to assist in negotiating each stage and then the specific areas of difficulty are revealed from the hints required and importantly all students reach a solution, and this reduces their anxiety when they approach a new problem So these are the sorts of hints then for conceptualizing the problem: read the question carefully, what are the important facts being given in representing the problem? Again the student may be able to work through a number of these steps with no help at all, and then they get to the point where they’ve… the unknown value but they don’t know what to do next, and so you might say well let’s build an equation starting with the pro numeral, or the letter that you’ve already generated, sometimes students get blocked when they’re trying to make this expression If the problem is quite complicated and you may need to give an analogy, this is still a representational skill but the analogy is relating it to perhaps themselves and their siblings, and this analogy hint will help contextualize the problem and help them make this step So in summary then, the research based instructional principles that we’ve covered in this video are to enhance number sense and that can be done through intensive number comparison or linking number to space through the number beads, for example, or it could be using the number line estimation task to instantiate the spatial array of numbers We need to submit this non symbolic symbolic link, repeated association of non symbolic and symbolic numbers will assist with that, encouraging increasing reliance on symbols and removal of the manipulative but making them available if needed still We need to conceptualize and automatise arithmetic, students can’t just apply procedures without having a conceptual understanding So we start with concrete representations and then we move to learned number facts from those, and of course we need to maximize motivation through positive reinforcement Reducing difficulty by adapting to the student’s needs and consistently referring to numbers in the environment, and last but not least, let us make it entertaining for them, let’s make it into a game, that will help them understand Now there are some workbooks there for you to access There are many websites for understanding learning difficulties I hope that some of what’s being presented though will give you at least some stepping stones to do to the remediation for students with maths learning difficulties Now that concludes this series of videos about maths learning difficulties and dyscalculics that were developed in this collaboration between the Victorian Department of Education and Training and the University of Melbourne If there are any references that I’ve made that you’d like to follow up or any further information you’d like based on our clinical experience of working with MLD students and dyscalculics please don’t hesitate to contact the Department directly or ourselves We are easily accessible through the university find an expert link and our email address is there and I’d like to thank you for your attention to the material I’ve presented today in these five videos over time, we hope you found them informative and useful