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Developing number and maths skills Joanna Nye and Gillian Bird Abstract - Most of us use numerical and mathematical skills on a day-to-day basis, even if this is only at a simple level of reading and writing numerals, often taking these skills for granted. It is perhaps surprising that very little research has been carried out investigating the development of these skills in children with Down syndrome. At the Centre we are aware of the increasing demand from parents and teachers for more information about these skills. This article presents a summary of the main research findings, starting with a brief overview of research carried out with typically developing children. A few practical ideas are also included. Keywords - Down Syndrome, Maths, Number, Children Introduction Most of us use numerical and mathematical skills on a day-to-day basis, even if this is only at a simple level of reading and writing numerals, often taking these skills for granted. Dealing with money, carrying out simple calculations, writing down telephone numbers and telling the time are all skills that are required by independent adults. It is perhaps surprising that very little research has been carried out investigating the development of these skills in children with Down syndrome. At the Centre we are aware of the increasing demand from parents and teachers for more information about these skills. This article presents a summary of the main research findings, starting with a brief overview of research carried out with typically developing children. A few practical ideas are also included. Research with typically developing children A large body of research has been carried out looking at the numerical skills of typically developing children without Down syndrome. The earliest skill that has been identified is the ability to distinguish between groups of different sizes at 5-months of age. Some think that this provides the first building block in developing numerical skills. The next major number milestone is the development of counting. A set of five principles (or sub-skills) are generally accepted as defining what counting consists of [1]. These are: 1. The one-to-one principle, which involves giving each item in an array a distinct ‘tag’, such that one and only one tag is used for each item. 2. The stable order principle, involving the use of a stable list of tags that is used for all sets counted. 3. The cardinal principle, where the final tag in the count represents the set as a whole. 3. The abstraction principle, for which the preceding principles can be applied to any array or collection of items. 4. The order-irrelevance principle, where the order and tag that each item receives is irrelevant. This principle is said to distinguish counting from labelling. Counting is thought to provide the foundations for later emerging arithmetical skills which many children find difficult to learn. Children are often good at simple calculations in meaningful contexts (for example, if we put these two soldiers with these three soldiers, there will be five soldiers altogether) but have difficulty in transfering their skills to more abstract problems, particularly the formal maths taught at school and using the language of this system [2]. The strategies that children use to solve arithmetical problems and how they choose between them have also been studied in some detail. Strategies include using physical materials to support calculations (e.g. counting on fingers), retrieval of facts from memory and the use of better known facts to infer answers. Children vary in the strategies they use first, but generally use more sophisticated and quicker methods (e.g. fact retrieval) as soon as they can rely on their accuracy. If you are interested in reading about any of these areas in more detail several books exist which provide excellent reviews [3] and [4]. In contrast to the amount we know about the standard development of number skills, very little is known about the same processes in children with Down syndrome. What can we expect from children with Down syndrome? Surveys of groups of children and case studies of individuals provide information about the skills of children and adults with Down syndrome. These describe skills attained, often relating them to specific interventions and can sometimes suggest useful advice but they do not give detailed insight into the processes that the children are using or how these develop. In 1987 Sue Buckley and Ben Sacks [5] published a survey of 90 families with a teenager with Down syndrome. The children had received little early intervention compared to children born since 1974. None attended mainstream schools. Only 18% of the sample were able to recite numbers or count objects beyond 1-20. Approximately half could do some simple addition. Few could do simple multiplication or division. Skills with money were also surveyed, one of the most useful everyday number skills, and only 6% were able to manage independently in a shop. In 1988 Janet Carr [6] presented data taken from a longitudinal study of 41 children with Down syndrome. At the age of 21 the average young adult's maths skills compared to those of a typical 5-year-old, whereas the average reading age compared to that of an 8-year-old. This pattern where maths lags behind reading ability has also been found by many other researchers [7, 8, 9, 10]. Can we predict the skills that a child with Down syndrome can achieve? In 1990 Sloper, Cunningham, Turner and Knussen [11] found a significant correlation between numerical ability and mental age. Numerical attainment of 117 children with Down syndrome was measured by a questionnaire (completed by the children’s teachers), assessing a range of competencies from “Discriminates between largest and smallest groups of objects” to “Does simple division work”. The children displayed a wide range of abilities, but skills were not reported in detail. Overall educational attainment was found to correlate with mental age and type of school attended, with mainstream schools being connected with favourable educational attainments. This was supported by Casey, Jones, Kugler and Watkins (1988) [12] who found similar benefits for children with Down syndrome attending mainstream schools, including in numerical ability. Baroody (1986) [13] tested 100 children with learning disabilities (not specifically children with Down syndrome) using a series of games investigating numerical skills which children are expected to have when entering school in the US. Baroody found wide individual differences, with some children with lower IQs in the younger age group performing at a higher level than some of the children with higher IQs in the older age group. Baroody concluded that labels such as IQ are not useful in predicting levels of numerical ability, conflicting with the conclusions drawn by Sloper et al. Differences in the way the studies were carried out may account for this discrepancy, but at least this highlights the need to look at an individual’s skills, rather than just using a label such as ‘Down syndrome’ or IQ scores to guide expectations. The different measures used by Sloper et al. and Baroody have both been used to test 16 children with Down syndrome aged 7 to 12.5 years in research based at the Sarah Duffen Centre [14]. The children displayed a wide range of abilities. No simple progression of skills could be seen, indicating wide variations in individual differences. Most of the children were able to carry out counting and simple addition up to 10 with concrete materials, and carry out simple sums with written numerals up to 10 with help. Two of the children were able to add and subtract numbers up to 20 without help. These were also the two oldest children in the group, suggesting that skills continue to improve with age and appropriate teaching, with the reasonable expectation that their skills will develop further through their teenage years and young adulthood. Are number skills improving? At present it is not clear whether the group of children with Down syndrome studied in 1995 have improved numeracy skills compared to those studied by Buckley and Sacks (1987) or Carr (1988). In 1994 Billie Shepperdson [15] compared the number skills of two generations of teenagers with Down syndrome. Professionals were asked to complete a questionnaire giving information about simple through to more difficult number skills, and the teenagers born in the seventies scored better than the sixties group both as teenagers and as adults. This supports the suggestion that educational opportunities have improved for people with Down syndrome, and as very few of the children in Shepperdson’s study attended mainstream school there is every reason to think that these have improved again for current and future school children with Down syndrome. Counting skills The only area where the processes underlying numerical skills have been investigated in children with Down syndrome is counting. In 1974 Cornwell [16] suggested that children with Down syndrome could only learn to count by rote, being unable to make use of the counting principles. This causes inflexibility in counting (for example, only being able to count red blocks laid out in a straight line), restricting the usefulness of the skill. More recent studies [17, 18] have provided examples of children with Down syndrome who are able to count flexibly and can therefore count objects in situations that they have not encountered before. For example, count objects laid out in a pattern that they have not encountered before, such as a circle, or being told to count so that the third item is tagged ‘one’, needing the child to ‘skip around’ the objects in order to count them all. Successful completion of such a task mean that a child is able to make use of all five of the counting principles. If a child make mistakes when counting then the particular errors they make can indicate which of the principles they do not make use of yet and what teaching should focus on next. This research demonstrates that children with Down syndrome can learn to count in a useful, flexible way, but they may not always do so. What can be done to improve children with Down syndrome’s number skills? Factors that are likely to influence development are improvements in language and memory skills and appropriate experience of counting. Exactly what these experiences are have yet to be determined, but we do have some ideas that may help your child. All of these ideas emphasise the need for variety and visual stimulation to keep your child motivated. The language of number The first language requirement for number work is to learn the sequence of number words. These can be learnt by rote and need not be ‘understood’. This understanding may develop later as more number activities are experienced. Once the count sequence has been mastered, children are initially unable to start the count string anywhere other than ‘one’. By experiencing starting counts at other numbers a child will be able to use the sequence more effectively for basic arithmetic [20]. Signing systems also have signs for number words and may help consolidate skills through multi-sensory learning. It should not be taken for granted that your child understands all the vocabulary involved in number work. Many words that are used may already be used in other contexts but have slightly different meanings when used with number. So some work may need to be done to develop this vocabulary and the related concepts when it is needed. Learning to read the word at the same time that the concept is learnt should help the child to remember the vocabulary. A mathematical vocabulary set can be best learnt in a hierarchy, for example starting with ‘size’, then ‘big’ and ‘small’, then ‘tall/short’, ‘long/short’, ‘wide/narrow’, etc. Any related concepts that the child already knows can be used to develop understanding. Visual aids are also the best way to teach these ideas, either pictures or real objects. Gestures can also be used to draw attention to the aspect of the object that is being looked at. The following is a list of vocabulary used in number work: * Size: width, height, length, big, small, little, fat, thin, long, short, thick, wide, narrow, biggest, smallest, longer, bigger, shorter than, longer than, as big as, order, compare. * Area, volume, capacity: a lot, lots, a little, a bit, a small bit, empty, full, much, most, more than, less than, same. Weight: heavy, not heavy, light, heaviest, lightest, heavier than, lighter than. * Units of measurement: grams, metres, etc. * Number and algebra: number words, a lot, all, some, both,another, not any, many, same, more, less, every, enough, as many as, first, second, third, etc., last, add, subtract, take away, guess, estimate, two times, multiply, units, tens, hundreds, repeating pattern, odd, even. * Fractions: same, different, as big as, smaller than, larger than, whole, piece of, part, complete, half, halves, equal, unequal, quarters, one quarter, two quarter, one half. * Time: again, now, after, soon, today, before, later, yesterday, early, late, once, tomorrow, twice, quick, slow, first, next, last, days of the week, weeks in a month, months in a year, time telling terminology, o’clock, half past, quarter to, quarter past, etc. * Money: coin names. * Shapes: round, dot, spot, line, circle, rectangle, hexagon, pentagon, square, oval, triangle, diamond, 3D shapes - sphere, cube, cylinder, cuboid, pyramid. * Properties of shapes: curved, rolls, flat, corner, edge, straight, right angle, turning, flip, symmetry, clockwise, anti-clockwise. * Spatial relationships: in, on, under, by, beside, behind, in front of, next to, over, through, inside, outside, out, to , off, above, below, round, up, down, front, back, left, right, forwards, backwards, top, bottom, middle, first, next, last. * Colours, material names, textures: wood, plastic, metal, etc. Rough, smooth, furry, etc. * Using and applying mathematics: results, outcome, check, explain, record, make, test, predict, prediction. * Handling data: sets, maps, diagrams, data collection, methods of recording data, for example, tables, lists, charts, graphs. Activities Counting and arithmetic are so useful to us because the abstract concepts about quantities can be applied to any situation. Therefore practising counting and arithmetic in a variety of contexts is likely to promote more flexible use. Repeated practice improves number skills, with skills becoming ‘automatic’ and facts being quickly retrieved from memory, both of which reduce the amount of effort that is required to complete a task and hence make it easier. This is where schemes such as Kumon Maths or the Montessori method may be of use in making number bonds and arithmetical facts easy to retrieve from memory. Teaching materials need to be varied with interesting presentation and application to keep children motivated. These can be made humorous or made ‘real’. From what we know about the learning patterns of children with Down syndrome [19], attention should be drawn to the connection between the procedures practised with abstract materials (e.g. counting blocks) and used in ‘real life’ (e.g. counting out cutlery when laying the table). Counting Arthur Baroody is an experienced researcher in children’s counting and has provided useful guidelines for remedying common difficulties that children experience [20]. He stresses the importance of careful accurate practice with an adult drawing the child’s attention to the relevant aspects of the task, and the use of games to aid motivation. Board and dice games are an excellent resource for practicing counting, and good fun too. The first skill that a child needs to develop when learning to count is pairing one tag with one item (the one-to-one principle). Before counting begins properly this concept can be aided by practising pairing sets of items together (e.g. lids and jars), both real and pictures. Once the number sequence has begun to be learnt then counting small sets of objects can be practiced. There then needs to be an emphasis on accuracy in pairing one number tag with one object, and making sure that each object has a tag. Also involved in one-to-one counting is keeping track of which items have and have not been counted. Using a finger to point can help, but care must be taken in coordinating points and number tags with objects. Other strategies can be used such as moving counted items into a separate pile or marking counted items on paper worksheets. Counting objects laid out in a line is easier to start with than objects scattered about. Once your child has developed some skill in this task, try practising counting objects laid out in different patterns (e.g. in a square, triangle, zig-zag) and randomly arranged, which require more advanced strategies for keeping track of counted items. Baroody suggests that a child should have some understanding of the one-to-one principle and count to 5 before work begins on the cardinality principle - that is understanding that “1, 2, 3, 4, 5 cars” means that there are “5 cars”. One tool for teaching this principle is the hidden-stars game. Using cards with small numbers (2-5) of stars on them, count the stars for the child “1, 2. There are 2 stars”. Cover up the card and ask “How many stars am I hiding?” If they do not answer correctly, the card can be revealed “Look, there are 2 stars”. Once your child is successful at this task then let them count the stars themselves before covering them up. This task can be continued, gradually adding in variations if you like, until they have a good grasp that counting tells us how many of something there is, rather than just being an activity in itself. Practice to learn the order-irrelevance principle can then begin. Objects can be counted and then moved about. The child is asked how many they think there are now, before counting to check their conclusion. Repeated counting of a set of items laid out in different patterns, along with discussion with an adult, will allow the child to realise that no matter what arrangement they are in, five cars are still five cars. Producing a requested amount The production of a specific number of items from a bigger group may also cause difficulties. The child needs to remember how many items are to be counted out, while co-ordinating the count sequence with each item produced. Often children will continue to count out all the items in a pile rather than stopping at the specified quantity. Before moving on to producing items you should make sure that your child is able to accurately count a given set. Once again accurate practice at producing a set of items should remedy further difficulties, and activities such as moving a counter on a board game provides excellent experience. If the child seems to be having difficulties in stopping at the correct item, then point out that they need to remember how many to give. You may find that you need to help your child to develop a strategy, such as rehearsing the number required several times before the count starts. Again production of small numbers should be started with, increasing the size of the set once accuracy has improved. Judging the equality or inequality of sets The next step is to use the counting skills to compare between sets. Judgements about equality and inequality of sets are often made on the basis of length by young children to begin with, but they need to learn that this is not always an accurate strategy. Sometimes placing items one-to-one next to each other can help to make equality judgements but this is not always practical. A more useful strategy is to use counting. Practice at counting and comparing sets should be carried out, with stress being placed on the cardinal value of each set being compared. Small sets that are equal should be started with, moving on to sets that are clearly unequal (1 vs. 6) and then sets are less obviously different (2 vs. 3). Number dominoes or lotto games could be useful here, especially ones that use sets in different arrangements. Related to this judgement skill is the concept of ‘conservation’. This refers to the understanding that as long as nothing has been added to a set, even if it has been moved about, the cardinal value of the set remains the same or has been ‘conserved’. Again practising counting, moving the arrangement of the items and counting again should help this understanding of conservation to develop. Judgements about more and less Although initially it is not important to ‘understand’ the number terms as they are learnt, for later skills such understanding is important. To make judgements about which set has more, it is necessary to know that five is bigger than three, etc. Children do not always develop this understanding themselves and therefore it may be necessary to teach this idea, as well as the concepts of bigger, smaller, less and more. Before this is attempted the child should be competent in counting and the count word sequence. Activities using a ‘staircase’ concept of numbers may be useful. For example making lines of objects or blocks (e.g. multilink), first one block, then two, then three, etc., drawing your child’s attention to how two is more than one, and how the lines of blocks get longer as the numbers increase. Number labels can be added to each ‘step’. Activities such as adding play people to a bus can also be used, adding one at a time and saying each time “two on the bus...one more...three on the bus”. Games can be played which use comparisons between two dice rolled or cards drawn to judge the ‘winner’. Once your child can make judgements based on counting concrete materials then they can progress to making judgements based on numerals. More advanced concepts Wendy Rinaldi, a speech and language therapist with 'ICAN', has found the following strategy highly effective for improving language and maths concept learning in school-aged children with language impairments. The activities were presented as small group activities enabling easy differentiation but some could be adapted to work with individual children (or whole classes). Also the general strategy of using different activities to repeatedly practice a concept can be implemented with one-to-one activities, and the level of tasks adjusted to suit the individual child’s needs. With each set of concepts that is being learnt each of the following 12 activities are worked through, starting with the most basic tasks and gradually getting more difficult. This system means that the concepts and language are well practiced and experienced in a wide variety of contexts. As well as improving flexible use of concepts, using a variety of activities helps to keep the child motivated. The tasks are designed to emphasise the use of visual presentation methods wherever possible, so are particularly appropriate for children with Down syndrome. The examples given relate to teaching 2D and 3D shapes. 1. Hands on! Introduction to the topic using real objects and the vocabulary used. Visual aids and gestures used. 2. Matching games. e.g. 2D to 3D, real object to picture. “Here is a square. Can you find its 3D partner?” 3. Finding games. e.g. “Can you find a cube?” from selection of objects. 4. Posters. Poster of spaceships, spiders, robots, etc. each with all its parts made out of a different shapes (a circle spaceship, a square spaceship, etc.) or a poster of all the things the class can think of that are square, etc. 5. Colouring in to direction. e.g. using worksheets filled with shapes - “Colour in all the squares.” “Colour in all the triangles in blue and all the circles in green.” 6. Right or wrong? - give a statement and child has to say whether it is right or wrong. e.g. “A cube is a solid, right or wrong?” 7. Board game. Make a simple board game, with every other square marked as a ‘question square’. When a child lands on a question square thay have to take a card and answer the question on it. e.g. “What is the 2D friend of the cube?” “Name a 3D shape.” 8. Guessing game. Each child has a card with a shape on it. They take turns in asking each other questions until they can guess what is on each others’ cards. 9. Card games. e.g. happy families, pairs, slow snap. Taskmaster make playing cards with one blank side that are useful for making your own games like these. 10. Name two things. An object (beanbag, ball, etc.) is passed around the group sat in a circle, while one person sits in the middle with his or her eyes closed. The child in the middle says “Stop” and asks a question or gives an instruction “Name two 3D shapes”. Whoever has the object has to answer and then swaps places to sit in the middle. 11. TV Programmes. Each child takes a turn in pretending to be an expert on the topic (e.g. 3D shapes) talking to the ‘audience’ . The children can also ask each other questions. 12. Detective game. One person picks an object out of a set that is in the middle of the circle and keeps it secret. The rest of the group has to ask questions gradually eliminating the rest of the items until they can work out what it is (e.g “Is it a solid?”, “Is it the 3D friend of the square?”). No doubt you can think of many more activities that could fit into this framework, making use of any special interests that your child has. References 1. Gelman, R. and Gallistel, C.R. (1978). The Child’s Understanding of Number. Cambridge, MA: Harvard University Press. 2. Hughes, M. (1986). Children and Number. Oxford: Blackwell. 3. Geary, D.C. (1994). Children’s Mathematical Development: Research and Practical Applications. Washington, DC: American Psychological Association. 4. Siegler, R.S. (1991). Children’s Thinking. Englewood Cliffs, NJ: Prentice-Hall. 5. Buckley, S. and Sacks, B. (1987). The Adolescent with Down Syndrome. Portsmouth: Portsmouth Polytechnic. 6. Carr, J. (1988). Six weeks to twenty-one years old: A longitudinal study of children with Down syndrome and their families. Journal of Child Psychology and Psychiatry, 29(4), 407-431. 7. Pototzky, C. and Grigg, A.E. (1942). A revision of the prognosis in mongolism. American Journal of Orthopsy-chiatry, 12, 503-510. 8. Dunsden, M.I., Carter, C.O. and Huntley, R.M.C. (1960). Upper end range of intelligence in mongolism. Lancet, i, 565-568. 9. Gibson, D. (1978). Down Syndrome: The Psychobiology of Mongolism. Cambridge: Cambridge University Press. 10. Buckley, S. (1985). Attaining basic educational skills: Reading, writing and number. In D. Lane and B. Stratford (eds.), Current Approaches to Down Syndrome (pp. 315-343). London: Holt, Rinehart and Winston. 11. Sloper, P., Cunningham, C., Turner, S. and Knussen, C. (1990). Factors relating to the academic attainments of children with Down syndrome. British Journal of Educational Psychology, 60, 284-298. 12. Casey, W., Jones, D., Kugler, B. and Watkins, B. (1988). Integration of Down syndrome children in the primary school: A longitudinal study of cognitive development and academic attainments. British Journal of Educational Psychology, 58, 279-286. 13. Baroody, A.J. (1986). Counting ability of moderately and mildly handicapped children. Education and Training of the Mentally Retarded, 21, 289-300. 14. Nye, J., Clibbens, J. and Bird, G. (1995). Numerical ability, general ability and language in children with Down syndrome. Down Syndrome: Research and Practice, 3(3), 92-102. Available Online: http://www.down-syndrome.net/library/periodicals/dsrp/03/3/092/ 15. Shepperdson, B. (1994). Attainments in reading and number of teenagers and young adults with Down syndrome. Down Syndrome: Research and Practice, 2(3), 97-101. Available Online: http://www.down-syndrome.net/library/periodicals/dsrp/02/3/097/ 16. Cornwell, A.C. (1974). Development of language, abstraction, and numerical concept formation in Down syndrome children. American Journal of Mental Deficiency, 79(2), 179-190. 17. Gelman, R. and Cohen, M. (1988). Qualitative differences in the way Down syndrome and normal children solve a novel counting problem. In L. Nadel (ed.), The Psychobiology of Down Syndrome. Cambridge, MA: MIT Press. 18. Caycho, L., Gunn, P. and Siegal, M. (1991). Counting by children with Down syndrome. American Journal on Mental Retardation, 95(5), 575-583. 19. Bird, G. and Buckley, S. (1994). Meeting the Educational Needs of Children with Down Syndrome. Portsmouth: The University of Portsmouth. Available Online: http://www.down-syndrome.net/library/books/meeting-ed-needs/ 20. Baroody, A.J. (1992). Remedying common counting difficulties. In J. Bideaud, C. Meljac and J.-P. Fischer (Eds.), Pathways to Number: Children’s Developing Numerical Abilities. Hove and London: LEA Addresses * Taskmaster Limited, Morris Road, Leicester, LE2 6BR. * ICAN, Barbican Citygate, 1-3 Dufferin Street, London, EC1Y 8NA (Tel: 0171 374 4422). More practical ideas on how to improve your child’s counting and arithmetical skills: * Bird, G. and Buckley, S. (1994) Meeting the Educational Needs of Children with Down Syndrome. Portsmouth: The University of Portsmouth. Available Online: http://www.down-syndrome.net/library/books/meeting-ed-needs/ * Broadley, I. (1991) Encouraging cognitive development: Part 1: Ideas for teaching. Part 2: Number skills. Portsmouth Down Syndrome Trust Newsletter, 1(4), 4-11 and 2(1), 7-9. (Reprint ref N3) * Green, R. and Laxon, V. (1978) Entering the World of Number. London: Thames and Hudson.