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1. Understanding math learning disability to guide math teaching Marie-Pascale No�l Catholic University of Louvain London, June 23

2. Studies with babies have shown that 6 month-old are able to discriminate between a collection of 8 items and a collection of 16 items, but not between one of 8 items and another of 12 items. Actually, they can discriminate two collections if they differ by a ratio of 1:2. By 9 months, babies are able to discriminate between a set of 16 and a set of 24 items. Thus, they can discriminate collections that differ by a ratio of 2:3. As adults, we can differentiate, without counting, collections differing by a ratio of 9:10. Studies with babies have shown that 6 month-old are able to discriminate between a collection of 8 items and a collection of 16 items, but not between one of 8 items and another of 12 items. Actually, they can discriminate two collections if they differ by a ratio of 1:2. By 9 months, babies are able to discriminate between a set of 16 and a set of 24 items. Thus, they can discriminate collections that differ by a ratio of 2:3. As adults, we can differentiate, without counting, collections differing by a ratio of 9:10.

3. Approximate and analogical number representation Increasing acuity over development Math learning disability is due to an impairment at that level (lower acuity) ? persistent, specific disorder of numerical competences and arithmetic skills development in children of normal intelligence Research has thus let to the hypothesis that we are born with an approximate number representation that gets more and more precise over development. Research has thus let to the hypothesis that we are born with an approximate number representation that gets more and more precise over development.

4. Approximate and analogical number representation Increasing acuity over development Math learning disability is due to an impairment at that level (lower acuity) ? persistent, specific disorder of numerical competences and arithmetic skills development in children of normal intelligence Based on this assumption, some authors have proposed that math learning disability might be due to an impairment at the level of that approximate number representation.MLD, is a specific and persistent math learning difficulties observed in some children of normal intelligence, Based on this assumption, some authors have proposed that math learning disability might be due to an impairment at the level of that approximate number representation.MLD, is a specific and persistent math learning difficulties observed in some children of normal intelligence,

5. Population children from age 3 to 6 MLD children (10 y.o.) control children (10 y.o.) 20 adults Recently, Piazza has shown that when you ask children to select the larger of two collections, without counting,, children, as they get older are able to discriminate between closer and closer collections. Yet, performance of children with MLD are much worse than control children of the same age. Recently, Piazza has shown that when you ask children to select the larger of two collections, without counting,, children, as they get older are able to discriminate between closer and closer collections. Yet, performance of children with MLD are much worse than control children of the same age.

6. 64 teenagers of 14 y.o. Assessment of the acuity of their number sense Record of their past school performance in math In unselected population: w and math performance Similarly, in an unselected population of 14 y.o. Teenagers, Halberda has shown that the measure of their ability to select between a collection of blue dots and a collection of yellow dots, the one that has more items. This number acuity is correlated with past performance of these children in math. Similarly, in an unselected population of 14 y.o. Teenagers, Halberda has shown that the measure of their ability to select between a collection of blue dots and a collection of yellow dots, the one that has more items. This number acuity is correlated with past performance of these children in math.

7. For instance, when considering their math record at 8 y.o., good performance in math is associated with the ability to discriminate two collections that are very close from one another. For instance, when considering their math record at 8 y.o., good performance in math is associated with the ability to discriminate two collections that are very close from one another.

8. Yet ... Population 45 children with MLD (7 y.o.) 45 control children (7 y.o.) Tasks Non-symbolic symbolic However, contradictory results exist in the literature. For instance, with Laurence Rousselle, we have tested 7 y.o. Children with MLD or with no learning disability in magnitude comparison tasks using collections or Arabic digits. However, contradictory results exist in the literature. For instance, with Laurence Rousselle, we have tested 7 y.o. Children with MLD or with no learning disability in magnitude comparison tasks using collections or Arabic digits.

9. We observed that children with MLD took more time than control children to compare the magnitude of two Arabic digits, but they were performing just the same as controls when they had to compare collections. These results thus indicate that children with MLD do not suffer from a deficit of the approximate number sense. We observed that children with MLD took more time than control children to compare the magnitude of two Arabic digits, but they were performing just the same as controls when they had to compare collections. These results thus indicate that children with MLD do not suffer from a deficit of the approximate number sense.

10. In unselected populations Population 6 y.o. (n=26), 7 y.o. (n=31), 8 y.o. (n=27) Magnitude comparison Comparison of two symbols (1 - 9) Comparison of two sets (1 - 9 items) Math performance Calculation score (Woodcock Johnson Math) Math fluency (+,-,x in 3�) Similarly, Holloway and Ansari tested unselected children of 6, 7 and 8 y.o. and asked them to select the bigger among two Arabic digits or collections. They also recorded their math performance using two tests. Similarly, Holloway and Ansari tested unselected children of 6, 7 and 8 y.o. and asked them to select the bigger among two Arabic digits or collections. They also recorded their math performance using two tests.

11. They observed that math performance was significantly correlated with performance in the symbolic tasks using Arabic digits, but was not with . performance in the non-symbolic task where they had to compare sets of items. Thus, again, these results indicate that math performance is not related to the ability of children to discriminate between two sets of items, a task tapping the approximate number sense. They observed that math performance was significantly correlated with performance in the symbolic tasks using Arabic digits, but was not with . performance in the non-symbolic task where they had to compare sets of items. Thus, again, these results indicate that math performance is not related to the ability of children to discriminate between two sets of items, a task tapping the approximate number sense.

12. How can we account for these results ? Let us take a developmental perspective How do we learn number symbols ? To account for these incoherent results, we decide to take a developmental perspective and examine what we know about the learning of numerical symbols in children and on the role of the analog magnitude R� in that process. To account for these incoherent results, we decide to take a developmental perspective and examine what we know about the learning of numerical symbols in children and on the role of the analog magnitude R� in that process.

13. Model 1 Number symbols are learned and mapped onto the approximate number R� According to a first model, children are born with an approximate number R� and number symbols such as number words and later on, Arabic digits, get their meaning by being mapped onto this approximate number R�. Accordingly, we could say that the first two studies I presented you argue in favour of an impairment of the approximate number R� in children with MLD while the two last argue in favour of a difficulty in accessing that R� from number symbols. According to a first model, children are born with an approximate number R� and number symbols such as number words and later on, Arabic digits, get their meaning by being mapped onto this approximate number R�. Accordingly, we could say that the first two studies I presented you argue in favour of an impairment of the approximate number R� in children with MLD while the two last argue in favour of a difficulty in accessing that R� from number symbols.

14. Model 2 It takes more than one year for children between the time they are able to recite the counting list and the time they are able to understand the cardinal value of the numerals in that count list (�give me�) children do not understand the meaning of one, two, three, four by referring to the approximate number R� They need to build a new R�: a R� of exact numbers. However, when we look at children�s development, we are impressed by their difficulties to learn the meaning of the number words. Indeed, It takes more than one year for children between the time they are able to recite the counting list and the time they are able to understand the cardinal value of the numerals in that count list. Such a comprehension is tested by the �give me task� for instance, in which the child is asked to give a certain number of items. According to Carey, these difficulties are due to the fact that children do not simply map number words to an already existing R� . Indeed, they can not learn the exact meaning of number words by using an approximate number R�. Rather, they have to built a new R�, a R� of exact numbers. However, when we look at children�s development, we are impressed by their difficulties to learn the meaning of the number words. Indeed, It takes more than one year for children between the time they are able to recite the counting list and the time they are able to understand the cardinal value of the numerals in that count list. Such a comprehension is tested by the �give me task� for instance, in which the child is asked to give a certain number of items. According to Carey, these difficulties are due to the fact that children do not simply map number words to an already existing R� . Indeed, they can not learn the exact meaning of number words by using an approximate number R�. Rather, they have to built a new R�, a R� of exact numbers.

15. Model 2 The representation of the words �one�, �two, �three, �four� is progressive and is based on the ability to represent precisely small sets on 1-4 items (object file system used in subitzing) For the other number words, the child needs to make the analogy between the order of the words in the counting list and the successive numbers that are related by the function �+1� Learning the meaning of the words �one�, �two, �three, �four� is progressive and is based on the ability to represent precisely small sets of items from one to four, what is called subitizing. For the other number words, the child needs to make the analogy between the order of the words in the counting list and the successive numbers that are related by the function �+1�. For instance, as 6 is just after 5 in the counting list, then the cardinal value of 6 corresponds to the cardinal value of 5 plus one. Learning the meaning of the words �one�, �two, �three, �four� is progressive and is based on the ability to represent precisely small sets of items from one to four, what is called subitizing. For the other number words, the child needs to make the analogy between the order of the words in the counting list and the successive numbers that are related by the function �+1�. For instance, as 6 is just after 5 in the counting list, then the cardinal value of 6 corresponds to the cardinal value of 5 plus one.

16. Model 2 For S. Carey, it is only after children have built their R� of exact number, which is, by the way, specifically human, that they start to connect this R� with the approximate number R�. For S. Carey, it is only after children have built their R� of exact number, which is, by the way, specifically human, that they start to connect this R� with the approximate number R�.

17. A developmental account of MLD ? These models lead us to take a developmental perspective on the results obtained in the literature and to consider the age at which children were tested. And then, a clear discrepancy appears between studies who have tested young children who had just a few years of math classes and those who were tested some years later. Indeed, all of them show difficulties when children with MLD have to compare the magnitude of two Arabic symbols, but when they have to compare the magnitude of two sets of items, the younger perform equally well that controls while the older perform more poorly. These models lead us to take a developmental perspective on the results obtained in the literature and to consider the age at which children were tested. And then, a clear discrepancy appears between studies who have tested young children who had just a few years of math classes and those who were tested some years later. Indeed, all of them show difficulties when children with MLD have to compare the magnitude of two Arabic symbols, but when they have to compare the magnitude of two sets of items, the younger perform equally well that controls while the older perform more poorly.

18. A developmental account First deficit are not seen in non-symbolic numerical tasks not due to a deficit of the approximate number magnitude system First deficits are seen in symbolic numerical tasks due to a difficulty in building an exact representation of natural numbers So, the first deficit observed in children with MLD is not seen in non-symbolic tasks. So their first deficit does not concern the approximate number magnitude R�. The first deficit that is observed in those children concerns the symbolic tasks. So, their first difficulty is in understanding the meaning of the number symbols. Their first deficit is in building an exact R� of natural numbers. So, the first deficit observed in children with MLD is not seen in non-symbolic tasks. So their first deficit does not concern the approximate number magnitude R�. The first deficit that is observed in those children concerns the symbolic tasks. So, their first difficulty is in understanding the meaning of the number symbols. Their first deficit is in building an exact R� of natural numbers.

19. Later: deficit on non-symbolic tasks as well Hypoth: connecting the exact number R� with the approximate magnitude system would increase the precision of the later This refinement process would be weaker in MLD children, thus leading to ? between MLD and control children Later on, a second deficit appears that concerns the non-symbolic tasks as well, so when they have to compare the magnitude of two sets. Later on, a second deficit appears that concerns the non-symbolic tasks as well, so when they have to compare the magnitude of two sets.

20. First deficit is on the symbolic tasks Later: deficit on non-symbolic tasks as well Hypoth: connecting the exact number R� with the approximate magnitude system would increase the precision of the later This refinement process would be weaker in MLD children, thus leading to ? between MLD and control children As this model proposes that the exact number R� gets connected to the approximate number R�, we might hypothesise that this coupling might increase the precision of the approximate number R�. Accordingly, the more the child would process numbers in an exact way by counting, by doing calculations and so on, the more the approximate number R� would increase in precision. As children with MLD have difficulties with number symbols and are not attracted by number processing, we might expect that this refinement process would be weaker in those children which, would lead after a while, to difference between children with MLD and control in tasks tapping the approximate number R�. As this model proposes that the exact number R� gets connected to the approximate number R�, we might hypothesise that this coupling might increase the precision of the approximate number R�. Accordingly, the more the child would process numbers in an exact way by counting, by doing calculations and so on, the more the approximate number R� would increase in precision. As children with MLD have difficulties with number symbols and are not attracted by number processing, we might expect that this refinement process would be weaker in those children which, would lead after a while, to difference between children with MLD and control in tasks tapping the approximate number R�.

21. Implications for school and education 1. creating a representation of exact natural numbers Building a comprehension of the cardinal value of the first number words One, two, three, four: sequentially By subitizing Link between the cardinal of a set given by subitizing and the last word in the counting procedure Working on the analogy between going further in the counting list and adding items in a set As some of you are in the field of education, I also wanted to take some time to see what these research on MLD might teach us regarding how we should guide children in their numerical development. According to our model, the first challenge of the child is to create a representation of exact natural numbers. As we know, this is a very gradual process that starts with the word one, then two, then three, then four. This should be done sequentially by connecting each of these number words with the subitizing, that is, our direct and exact apprehension of small sets of 1 to 4 items. Then a link should be made between the cardinal of a set given by subitizing and the last word obtained in a counting procedure. This should help the child understanding the use of the counting procedure and the cardinal principle, that is, the fact that the last word in a counting process is the cardinal of the set. Then children should be helped in building an analogy between going further in the count list and adding items to a set. As some of you are in the field of education, I also wanted to take some time to see what these research on MLD might teach us regarding how we should guide children in their numerical development. According to our model, the first challenge of the child is to create a representation of exact natural numbers. As we know, this is a very gradual process that starts with the word one, then two, then three, then four. This should be done sequentially by connecting each of these number words with the subitizing, that is, our direct and exact apprehension of small sets of 1 to 4 items. Then a link should be made between the cardinal of a set given by subitizing and the last word obtained in a counting procedure. This should help the child understanding the use of the counting procedure and the cardinal principle, that is, the fact that the last word in a counting process is the cardinal of the set. Then children should be helped in building an analogy between going further in the count list and adding items to a set.

22. 2. Favouring the connection between the exact representation of natural numbers and the analogical representation of number magnitude Guessing and counting Using space to represent number magnitude The second challenge of the child is then to connect this R� of exact numbers with the old approximate number magnitude. This could be done by showing sets of items and asking the child to guess how many they are and then to count them or, conversely, by asking the child to grasp at once a certain number of items and then count to see if the guess was close or not. Another way of making this connection is by using the representation of space. Several intervention programs have been using spatial number lines. The second challenge of the child is then to connect this R� of exact numbers with the old approximate number magnitude. This could be done by showing sets of items and asking the child to guess how many they are and then to count them or, conversely, by asking the child to grasp at once a certain number of items and then count to see if the guess was close or not. Another way of making this connection is by using the representation of space. Several intervention programs have been using spatial number lines.

23. The number race Associating digits, collections and space In comparison, additions and subtraction For instance, Wilson in the team of Dehaene has developed the number race game which favours the associations between Arabic digits, collections and position on a spatial number line. This is done in comparison tasks as well as in additions and subtractions. They showed, in nine children with MLD, that playing this game for 20 hours increased significantly their performance in several numerical tasks. For instance, Wilson in the team of Dehaene has developed the number race game which favours the associations between Arabic digits, collections and position on a spatial number line. This is done in comparison tasks as well as in additions and subtractions. They showed, in nine children with MLD, that playing this game for 20 hours increased significantly their performance in several numerical tasks.

24. Faster in dots enumeration Increased accuracy in subtraction (but not addition) Increased accuracy & speed in magnitude comparison

25. Number board game Colour game Number board game Very powerful results were obtained by Ramani and Siegler who used the number board game. This is a very simple game in which the child spin a spinner and then moves its token to either the color indicated by the spinner and saying out loud the colour names of the squares, for instance, saying red, �orange,�, or moves the token by the number of steps indicated by the spinner, and saying the number of each squares, for instance, �four, five�. Playing the number games for less than two hours increased significantly the performance of low-income children in several numerical tasks, which was not the case for children playing the colour game. Very powerful results were obtained by Ramani and Siegler who used the number board game. This is a very simple game in which the child spin a spinner and then moves its token to either the color indicated by the spinner and saying out loud the colour names of the squares, for instance, saying red, �orange,�, or moves the token by the number of steps indicated by the spinner, and saying the number of each squares, for instance, �four, five�. Playing the number games for less than two hours increased significantly the performance of low-income children in several numerical tasks, which was not the case for children playing the colour game.

26. For instance, they increase their accuracy in a magnitude compariosn task and in a counting task. For instance, they increase their accuracy in a magnitude compariosn task and in a counting task.

27. The estimator For older children, another interesting tool has been developed by Bruno Vilette. In the estimator, the child first selects a game, either positionning number on the number line, or positionning the solution of an addition or of a subtraction. In the case of calculation, the child first reads aloud the problem, then, using the cursor, the child indicated the position of the approximate answer on the number line. If this approximation is far from the correct answer, the number corresponding to the marked position appears in red and the child has to restart its estimation. If the child�s approximation is close to the correct answer, the answer appears in green.For older children, another interesting tool has been developed by Bruno Vilette. In the estimator, the child first selects a game, either positionning number on the number line, or positionning the solution of an addition or of a subtraction. In the case of calculation, the child first reads aloud the problem, then, using the cursor, the child indicated the position of the approximate answer on the number line. If this approximation is far from the correct answer, the number corresponding to the marked position appears in red and the child has to restart its estimation. If the child�s approximation is close to the correct answer, the answer appears in green.

28. The estimator This tool has been used with 10 children with MLD. Playing this game for less than four hours significantly improved their performance in calculation and in a global math test (Zareki). More importantly, this game was more efficient than another computer game using the same arithmetical operations but focusing on exact solving procedures only. These types of new tools are very promising and are worth considering by educationists. This tool has been used with 10 children with MLD. Playing this game for less than four hours significantly improved their performance in calculation and in a global math test (Zareki). More importantly, this game was more efficient than another computer game using the same arithmetical operations but focusing on exact solving procedures only. These types of new tools are very promising and are worth considering by educationists.

29. 1. Building a representation of exact numbers 2. Connecting this representation with the approximate number representation

30. Thanks for your attention !

31. Adult study 22 adults who were MLD (24�9 y.o.) 22 adults control, same age, same studies or profession