Understanding the development of number and mathematics skills

[TODO: summary] …

number-overview

Introduction

There is very little research into the development of number skills in children, teenagers or adults with Down syndrome. A small number of papers have investigated early counting skills^[1–8] and others describe the achievements of small groups of teenagers or adults.[TODO: 9-13] Only four papers look at more advanced mathematical understanding of length, quantity, money and algebra.[TODO: 14-17]

Several studies identify that the number skills of pupils with Down syndrome are improving with better educational opportunities^[9],[TODO: 19] and that they improve with education in mainstream settings.[TODO: 4],[TODO: 10],[TODO: 12],[TODO: 13],[TODO: 7],[TODO: 18],[TODO: 20],[TODO: 21] Several studies suggest that reading skills are more advanced than numeracy skills for many children with Down syndrome but they do not explore the possible reasons for this.[TODO: 7],[TODO: 8],[TODO: 22],[TODO: 23]

The published literature offers little specific practical guidance to parents or teachers at the present time, though several papers highlight the benefits of systems that represent number visually,^[6] [TODO: 24-28] however, practical implications for teaching are identified in this review where possible.

The review begins with an outline of the development of number skills and knowledge as they are understood from research with typically developing children and then reviews the research with children with Down syndrome.

The significance of knowledge about the specific learning strengths and weaknesses usually experienced by children with Down syndrome is also taken account of in identifying teaching approaches that are likely to be helpful. In addition, we have included information based on their experience of working with children with Down syndrome in mainstream classrooms since 1988.

Learning about number and mathematics

The maths curriculum

The central part of the mathematics curriculum in school concerns learning to understand and use numbers for counting and for calculation. This is not a simple task for any child and to reach the stage of multiplication and division of large numbers takes some 6 years, usually from 2 to 8 or 9 years of age.^[10–12]. However, the mathematics curriculum also includes learning to understand money, time, shape, size and the measurement of quantities (length, height, weight, volume). Later on the curriculum includes geometry, algebra and the understanding of more advanced mathematical concepts, reasoning and calculations. The later part of the mathematics curriculum may be of less relevance for everyday life but the earlier parts of the curriculum do have important applications in the daily lives of most adults.

The foundations for understanding number, size, shape, quantity, money and time are being laid during the preschool years. The words for counting and for other relevant concepts are being learned during daily activities and games.

A number of authors have emphasised the social nature of number learning^[13–17] and the importance of practice [TODO: 30],[TODO: 45] to consolidate and extend children’s learning. The count words are learned in imitation as adults and children play counting games together and, for preschoolers, the first reason for counting things is because someone asks them ‘How many have you got?’ Therefore, parents and other adults play a very important role in teaching children about number and in setting up opportunities for them to practice their emerging skills.

The number system

Understanding and using the number system is more difficult than may be apparent at first sight.^[10] [TODO: 31] In order to be able to count correctly, children have to master several skills.[TODO: 38-41] First, children have to learn the names of the number words (the count word sequence) in the correct order (this is referred to as the stable order). Most typically developing 3 to 4 year olds know the number words from 1 to 10 in the correct order. Secondly, children have to learn how to count a small number of items so that each item is given a number name and only counted once (this is called ‘one-to-one correspondence’). Thirdly children have to understand that the last number they say in the counting task has a special meaning. It is called the cardinal number and it represents the total number of the counted items and is the answer to the question ‘’How many are there?’’ When children can correctly answer the ‘’How many?’’ question, they are described as having achieved cardinality. At this stage they can also respond correctly when asked to give a certain number of items from a larger set, for example correctly give 2 or 4 items from a set of 8. As they gain experience with counting activities, the nature of number becomes better understood and children realise that numbers represent quantity, that you can count all kinds of different items (the abstraction principle) and that the order in which items are counted is irrelevant so long as each item is only counted once (the order irrelevance principle).

Children also have to learn the written digits for the number words that they have learned to say by repetition and this becomes more difficult for the numbers from 11 to 19 as in English they do not follow a logical naming system. From 20, the system becomes logical in its representation of tens and units, as we say ‘twenty-one, twenty-two, twenty-three’. In some languages (e.g. Chinese, Japanese, Korean) the same regularity for number names applies to ‘ten-one’, ‘ten-two’, ‘ten-three’, but this is not the case in English and the ‘teen’ words often confuse children. There is some evidence that children in countries with a regular number naming system for 11-19 learn to understand tens and units (place value) and calculate with the system more easily than English and American children.^[11]

Progress in pre-school years for typically developing children

At 2 to 3 years, typically developing children begin to use number words to ‘count’ as they play, showing that they are beginning to explore and understand counting. Many 3 year olds are beginning to correctly label small sets of 2 and 3 items, but will make mistakes with larger sets of items. The difference between small sets of 2, 3 and possibly 4 items are thought to be recognised by infants before they can count, and these are called ‘subitizable’ sets.^[18] Practice at counting these ‘subitizable’ set sizes has been shown to help children to understand cardinality and to then transfer this understanding to larger set sizes.[TODO: 43]

! Cuisenaire rods

Figure 1. Cuisenaire rods

By 4 years of age, many children show cardinality and can correctly respond to ‘’How many?’’ and ‘’Give me …’’ questions for numbers up to 10.^[10],[TODO: 44] They understand ‘more’ and ‘less’ for small numbers, i.e. that 4 is more than 3, but they do not yet understand the ordinal nature of the number system, i.e. that each ‘next’ number represents ‘one more’. The use of teaching materials such as Cuisenaire rods [TODO: 47] ([Figure 1]) and Numicon shapes [TODO: 29],[TODO: 48] ([Figure 2]) helps children to see that each ‘next’ number is one more and also that 2 and 2 or 1 and 3 are the same as 4.[TODO: 28],[TODO: 29]

Teachers of mathematics have differing views on the importance of counting activities, as some argue that counting activities alone will not lead children to understand the ordinal nature of the number system.^[19] However, others stress the importance of overlearning or procedural mastery of skills in cognitive development, arguing that mastery is needed at each level before newly acquired information becomes available for the next level of analysis. [TODO: 30],[TODO: 45] For example, the ‘count sequence’ needs to be ‘overlearned,’ in order, from one to ten, before it will be possible for the child to separate out the numbers in the sequence and count backwards and forwards from different numbers within this range, or to ‘count on’ when adding (i.e. if asked to add 2 items to a group of 4, the child can ‘count on’ from 4 i.e. ‘’4, 5, 6’‘, and does not need to start counting from 1 in order to carry out the task). A further benefit of overlearning by rote practice is that it leads to automatization of the procedure or skill. When a skill is so well mastered that it is ’automatized’ it requires little conscious effort to use, and therefore frees up working memory resources to be used for necessary mental processing during a task.[TODO: 49]

! Numicon shapes and materials ! Numicon shapes and materials ! Numicon shapes and materials

Figure 2. Numicon shapes and materials

Using counting knowledge to share begins at 4 to 5 years, using ‘’One for me, one for you’’ until the items are shared equally. Before this stage, children will give a handful to each child and not check to see if the piles are equal. At the next stage, children give a handful to each child and then count each pile. It is only at about 8 to 9 years of age that children count all the items to be shared and divide this number by the number of children to decide how many each child should have before beginning any distribution.^[10]

Progress in primary school years for typically developing children

Progress will vary among children at any age but the following guide to achievements is based on research studies^[10],[TODO: 31] and on the targets most children are expected to achieve, for their age, in the English school system. [TODO: 46]

Between 5 and 6 years, typically developing children begin to understand more about number concepts and the number system. They can recite the number words to 5 and then to 10, 20 or beyond, in familiar games and rhymes. They can recognise the written numerals 1 to 9 and count reliably up to 10 everyday objects. They can use words such as more or less, greater or smaller, heavier or lighter and begin to use the vocabulary involved in adding or subtracting, while combining groups of objects for adding or removing objects to ‘take away’. They will also be able to recognise and recreate simple repeating patterns (e.g. red brick, yellow brick, red brick, yellow brick… or two green pegs, one red peg, two green pegs, one red peg….). They can use words such as circle, square, triangle, cube, sphere, bigger and smaller to describe the shape and size of solid and flat shapes as well as everyday words for position, direction and movement. They begin to understand and use the vocabulary for time and to sequence familiar events. They learn the days of the week in order and begin to read the ‘hour’ time on the clock face. They also begin to understand and use the vocabulary related to money. They can sort coins and use them in ‘playing shops’.

Between 6 and 7 years of age most typically developing children can recite the number words to 20 and beyond, from and back to zero, and reliably count at least 20 objects. They can read, write and order numbers from 0 to at least 20, and begin to identify tens and units in the ‘teen’ numbers. They can also count on or count back in ones from small numbers and count in tens from zero. They will begin to count on and back from zero to 20 in twos and in fives. They will continue to develop their understanding of addition and subtraction and begin to use the signs +, - , = (operands). They will be able to ‘count on’ from the larger number when adding two numbers. They will learn how to double the numbers 1 to 5. They will be able to compare two lengths, volumes or capacities and develop further their knowledge of 2 and 3 dimensional shapes to include rectangle, cylinder, and cone. They will know the days of the week and the seasons of the year and read the time to the hour or half hour on an analogue clock. They will recognise coins of different values, find totals and change up to 20 pence and work out how to pay exact sums using smaller coins.

Between 7 and 8 years, most typically developing children can count, read, write and order whole numbers to at least 100 and know what each digit represents (hundreds, tens, units with 0 as a place holder). They know which are odd and even numbers and can count on and back in ones and tens. They understand that subtraction is the inverse of addition and that multiplication is repeated addition. They know the 2 and 10 ‘times’ tables. They can use a ruler for measuring and drawing lines, and use simple scales and measures for weight and volume.

At 8 to 9 years, most children know numbers to 1000 and can count on and back in tens and hundreds. Within 0 to 100, they can count on or back in twos from any two digit number and order numbers to at least 1000, on a number line or number square. They know the 2, 5 and 10 times tables and understand that division is the inverse of multiplication. At this age, they can use and begin to read the units of time (second, minute, hour, day, week, month, year) and know the relationships between them. They know the months of the year and can read the time to the quarter hour on an analogue clock. They can measure, weigh and compare lengths, masses and capacities using standard units. They can also use the abbreviated notations, ‘£’ and ‘p’, for money. They can identify right angles and make simple tables and graphs of data. This level of achievement would certainly provide the knowledge and skills necessary for most everyday life and work situations requiring number or maths skills.

The wide range of typical development

While the previous sections are based on the average development of children in each age group, many typically developing children struggle with basic numeracy skills. It is currently estimated that 25% of the adult population of the UK do not reach the number skills level of an average eleven year old, and will have difficulty calculating change in shops.^[20]. Many of these adults will also have limited literacy skills. This indicates that there are many non-disabled pupils in mainstream schools making slow or little progress, even at secondary level (11-16 years).

What influences the progress in number and maths of typically developing children?

The following section identifies a variety of factors that will influence all children’s learning and which have relevance for understanding the difficulties of children with Down syndrome. The development of children’s number skills have not received the same amount of attention from researchers as the development of children’s language skills, but there are some useful findings.

Research into infants’ knowledge and the counting abilities of animals^[10] supports the view that number skills are based on an innate system that is at least partially independent of other biological systems, such as those underlying language acquisition.[TODO: 50],[TODO: 51] This could have relevance for children with Down syndrome, as it should not be assumed that their documented language difficulties necessarily mean that number skills are equally impaired. Some children and adults with significant language and cognitive delays (often with autistic spectrum disorders) can be exceptionally able at number, supporting the view of number as an independent cognitive skill. Some authors have argued that the principles underlying counting may be innate,[TODO: 38],[TODO: 41],[TODO: 55],[TODO: 56] but others have argued that children learn them by induction as they experience a wide range of counting activities.[TODO: 39],[TODO: 40]

Research with typically developing children indicates that number progress is influenced by:

• Social experiences and exposure to number in preschool years
  Children’s opportunities to begin to learn about number and maths concepts begins in pre-school years and therefore children’s knowledge when they start school will vary according to the quality and extent of their learning opportunities at home and at preschool. Parents should be encouraged to draw children’s attention to the uses of number in their everyday lives and to engage them in games to teach them about counting and quantity.
• Teaching methods, use of practical materials to support understanding of number relationships, importance of practice and rote learning of basics
  In the English numeracy curriculum there is a heavy emphasis on counting in the early years and on mastering mental arithmetic. Teachers of mathematics have differing views about the best ways to teach children to count and to understand the number system. Some argue that counting activities alone will not lead children to understand the nature of the number system and advocate the use of materials which provide a visual spatial representation of the system (such as Numicon and Cuisenaire) to help children.^[19] Numicon has been demonstrated to improve the maths progress of typically developing primary school children in the school where the materials and activities were developed. In the Numicon approach, a wide range of counting activities are advocated but the Numicon materials and activities have also been designed to support the development of mental imagery for whole numbers, which in turn will support mental arithmetic.
  Drill and practice tends to be unfashionable but there are good arguments for suggesting that children should practise the count word sequence until it is mastered to an ‘automatic’ level, and similarly learn multiplication tables and other useful addition skills (e.g. adding all combinations of 2 numbers, for 1 to 9, adding in 10’s, 5’s and 2’s), so that they do not have to be consciously calculated when needed. Automatization of skills frees up space in working memory - the mental workspace used for calculations and problem solving. [52-54]
  A combination of a wide variety of counting and quantity experiences, the use of a visual image system to illustrate the ordinal nature of the system, place value and the relationships between numbers, and rote practice of number words, procedures for calculations and number facts, is probably the best approach.

• Knowing the language and concepts for maths
  Children’s vocabulary when they start school needs to include the words for number and maths that they will need in the classroom, though beyond this, it is not clear how language and number abilities link. Parents play an important role in ensuring that they draw their children’s attention to size, colour, shape and other attributes during every day experiences and introduce them to comparing and contrasting activities. Children with speech and language impairment often have difficulty with number and maths in school^[21] but the reasons could be both limited relevant vocabulary and working memory difficulties, as these are usually associated with speech and language impairment. Working memory capacity and function is important for maths and all mental activities. It is discussed further below and explained in detail in a separate module.
• The relevance of the skills to everyday life
  Researchers have shown that children and adults can develop a high level of skill in calculating outside of formal school systems (e.g. the money calculating skills of street children and illiterate adults, the ability to score at games) but are not always able to transfer these abilities to the formal maths of the classroom.^[11] Conversely, they have also shown the positive effects of using examples and materials in the classroom that show children the relevant applications of their maths learning in their everyday lives.[TODO: 62] Many children might be more interested in mastering the basics of counting and calculating to ‘hundreds’, tens and units if they realised its main purpose in their adult lives will be when spending money.

• Reading ability
  The ability to solve maths problems that are presented as written problems (e.g. ‘’Jenny has three 3 marbles, Bill gives her 2 more. How many has she got now?’‘) is influenced by reading ability and the ability to inference from text. Most children are slower at solving problems presented in this way [see^[10] and initially need help to translate them into a number problem in order to identify the appropriate calculation. For example, the problem’‘Jenny has 3 marbles. She has one less marble than Mary. How many does Mary have?’’ is more difficult for children than it may seem as the keyword ‘’less’’ in the problem often leads them to think that they have to do a subtraction, rather than an addition.

• Motor skills for counting and recording (writing numerals)
  Children’s early progress with counting will be influenced by their motor dexterity as they use bricks or other objects to support counting at home and in the classroom. Their ability to write the numerals and to record their work will influence their progress and some teachers recommend that children are not asked to write down numbers in their first year in school, as this may distract from activities that will help them to understand the number system. Strategies to help children and teenagers with particular writing difficulties could include numeral cards so that they can record the steps in a calculation using these cards instead of writing. Computer programmes will also help as they may enable the child to choose the answer using the mouse or use of number keys.
• Working memory capacity
  Children in regular classrooms with poor working memory skills for their age have been shown to have difficulties in number and maths.^[22],[TODO: 59] Working memory can be defined as the ‘mental workspace’ which supports the processing of incoming visual and auditory information, and the conscious processing and manipulation of this information. The amount of information that children can hold and process in this system in a limited time increases with age. Children who do not show age-appropriate development of working memory skills will have difficulty learning new information in the classroom and they will have difficulty with tasks which require them to perform some kind of mental operation on information - such as doing mental arithmetic or reading for meaning. These types of tasks are made up of sub-skills, e.g. mental arithmetic will be based on a child’s knowledge of numbers and number facts, and if these are available for automatic recall this will reduce the load on the working memory system. Similarly, if word recognition is automated in the reading task, and therefore the child does not have to stop to work out words in the text, more working memory capacity can be devoted to understanding the text. Children and teenagers with working memory difficulties can be helped by the use of visual supports for learning whenever possible as these will reduce the load on the memory system. They will also be helped by the overlearning and ‘automatizing’ of basic facts and procedures.[TODO: 52-54]
• Logical reasoning ability
  The number system is a logical system and the ability to reason logically and work out relationships by inference will help children to understand the system, carry out calculations and solve problems.^[11] However, many children who find logical reasoning difficult can learn to understand and carry out a range of calculations by using ‘rules’ (i.e. learned procedures) and therefore acquire useful functional number skills.

What do we know about the number and maths skills of children with Down syndrome?

Information on the progress of children with Down syndrome with number and maths is very limited and, as with reading and writing, the maths instruction offered to many children with Down syndrome in the past would have been minimal. This means that the surveys of achievements of teenagers and adults in the existing literature should not be taken as a guide to the potential of individuals with Down syndrome.

! Work by Joni, aged 10 years

Figure 3. Work by Joni, aged 10 years.

In our current experience, based on supporting children and teenagers with Down syndrome in mainstream schools for the past twelve years and assessing many other children and teenagers each year, there is wide variation in number ability among individuals with Down syndrome. They have worked with ten year olds with Down syndrome who are able to add, subtract, multiply and divide 3 digit numbers and add fractions, and are keeping up with their non-disabled peers. They have also worked with teenagers who are still struggling to learn read, write and count numbers to 20. Typically, pupils’ achievements in number are at a lower level than achievements in literacy.

! Work of a 13-year old Italian pupil

Figure 4. Work of a 13-year old Italian pupil.^[9] (Reproduced with permission)

The reality is that the potential maths abilities of individuals with Down syndrome are not yet known, nor do we have enough information to understand the wide variation in the achievements of adults at the present time. Martinez^[23] working in Italy has provided examples of teenagers with Down syndrome who are able to do fractions and algebra.

It is also likely that many adults with Down syndrome could improve their number and maths skills with appropriate teaching. In a recent report, also by Martinez, a man of 51 years is successfully learning to count having missed out on earlier educational opportunities.^[9] As his work in [Figure 5] shows, he demonstrated considerable drawing ability as he worked on his number work.

! Learning to count at 51 years of age ! Learning to count at 51 years of age

Figure 5. Learning to count at 51 years of age.^[9] (Reproduced with permission)

The findings of research studies

Research evidence has come from projects that measure or evaluate individuals’ skills and development in different ways, including surveys of achievements, experimental studies, studies evaluating particular teaching methods and single case studies. Most studies have looked at school age children, teenagers or young adults. Only two studies have looked at the development of number skills in the preschool years.

Surveys of achievements

Surveys of the skills of groups of individuals with Down syndrome within certain age ranges provide some limited information about the range of achievements, but this needs to be interpreted in the light of their educational experience when the surveys were undertaken.

One UK study published in 1988 which looked at the progress of teenagers or young adults with Down syndrome, born in 1963/64, reports very limited achievements. [TODO: references 60] In this longitudinal study, two thirds of the 45 sixteen year olds could recognise numbers and count, but less than half could add single numbers, less than a quarter could do subtraction and only 2 could multiply or divide. At twenty one years of age, more than half could count but only two could handle addition or subtraction of 2 digit numbers.

It is important to recognise that the young people in this survey would have received limited education as they did not have a right to education until 1971, therefore these findings tell us little about the potential number and money abilities of individuals with Down syndrome. Some evidence is accumulating to support the view that their potential is higher than has been assumed, as better education is leading to reports of higher achievements in a number of countries. Studies from the USA,^[24] Australia [TODO: 10] and the UK [TODO: 12],[TODO: 13] all report advances in the levels reached.

The UK study of 46 teenagers with Down syndrome^[25] [TODO: 12,13] has recorded numeracy achievements for pupils of similar ability, aged 11-18, in 1999. These pupils have been educated in either special or mainstream educational placements in one county in southern England. The authors carried out a similar study of teenagers in the same area in 1987,[TODO: 11] so they have been able to compare the results of the two cohorts and look at the effects of inclusive education on outcomes. The mean ages of the groups are 14 years 1 month for the 1987 group, 16 years 4 months for the 1999 special school group and 14 years 8 months for the 1999 mainstreamed group. This mean age difference could not be avoided in collecting the 1999 sample and is the result of changing policies on inclusion over time. In the study area, a larger proportion of the older than the younger teenagers were in special school. The data on number, time and money skills from these studies is set out in [Table 1] and [Table 2]

Table 1. The achievements of teenagers in 1987 and 1999, in special or mainstream schools (giving the percentage in each group achieving the skills): Time and money skills

Table 2. The achievements of teenagers in 1987 and 1999, in special or mainstream schools (giving the percentage in each group achieving the skills): Number skills

There are two striking findings in relation to the teenager’s achievements in number and time.

Firstly, the teenagers in mainstream schools have reached a significantly higher level of achievement than their peers of similar abilities in special schools.

Secondly, there has been almost no improvement in the achievements of pupils in the special schools over the decade, from 1987 to 1999.

The pupils in the special schools do have better money skills at the time of the survey. This may be because the schools have been more effective in teaching money but also because the pupils in the special schools were on average almost two years older than the mainstream group and may have had more experience of using money in shops or restaurants. The fact that the mean age of the special school group in 1999 was two years older than the mainstreamed group makes the differences of the achievements of the mainstreamed group on all measures except money even more significant.

The authors believe that the gains in the mainstream schools are largely due to learning maths in classrooms with a competent peer group and being included in the regular curriculum lessons with individual support to differentiate that curriculum. The mainstreamed teenagers have probably also had more intensive teaching and practise opportunities, with an hour each day currently being given to numeracy teaching in UK schools. They also believe that the teachers in the mainstream schools have higher expectations of the pupils’ performance.

Another UK study provides some information on number progress and possible links with language and literacy progress. In a longitudinal study undertaken by Byrne (1997) with 24 pupils with Down syndrome attending mainstream primary schools, the children’s progress was measured over a two year period, from 1995 to 1997. Although the study was primarily investigating the development of reading skills, standardised measures of number ability were also used and the pupil’s results recorded. Each of the pupils with Down syndrome received a high level of individual learning support in the classroom. For this group of pupils, aged between 6 years and 14 years at the end of the study, language and number skills were approximately 4 years behind the pupils mean chronological age, whereas reading accuracy and spelling ages were only 2 years behind.

[Figure 6] illustrates the relationships between the skills measured in the study and the way that they progressed over time. The children were divided into 3 groups on the basis of their reading abilities. The alphabetic group were showing the ability to decode words using their phonic knowledge, the logographic group were reading using sight-word recall predominantly and the third group were only at the beginning stages of independent reading. However, each group clearly shows the same relationship between number skills and language comprehension measures and the same relationship between their number skills and reading skills. The study is not able to offer any explanation for the better reading than number skills, a finding reported by other authors.^[7,8,26] [TODO: 60]

!

Figure 6. The relative levels of attainment and progress made by the Alphabetic Readers (n=7), Logographic Readers (n=9) and Nonreaders (n=8) in the Down syndrome group.^[27] (Reproduced with permission)

Similarities and differences for matched samples

Studies that compare children with Down syndrome with other children usually do so to investigate the way in which the pupils learn. If they identify any differences in learning ability or style associated with having Down syndrome, these differences may have implications for teaching and learning. Comparison groups can be pupils who have developmental delay but do not have Down syndrome or pupils who typically developing. Matched groups of typically developing children will be significantly younger than pupils with Down syndrome and their life experience will be different, making results sometimes difficult to interpret.

Studies of infants and pre-school children

One group of researchers have recently investigated the awareness of quantity by seeing if infants with Down syndrome can ‘see’ subitizable sets at the stage that other infants can. Paterson^[28] looked at a group of infants with Down syndrome at 30 months of age and they did not show the same ability to ‘see’ the different set sizes as non-disabled, or Williams syndrome infants of the same mental age (15-16 months). This is the only study of such early understanding and discrimination skills in infants and needs to be replicated. However, the significance of the findings are difficult to interpret as some authors point out that the links between these early infant skills and later understanding of number are yet to be established.[TODO: 31] Indeed, the Paterson study includes evidence that adults with Down syndrome are better at number than adults with Williams syndrome, the reverse of the infant’s performance.

Another study by Nye just being completed in the UK^[5] at the start of the study, over a two year period. It has recorded their progress in mastering early counting skills and cardinality, compared with a group of typically developing pre-schoolers matched with them for non-verbal mental age. The first data is about to be published and when all the data is available it should provide information on the range of development seen over the two years and the extent to which progress is linked with either language or non-verbal reasoning abilities of the children. The effects of parental strategies when they support their children’s counting is also being documented, as parental help plays a significant role in helping children to understand counting and number. Preliminary data from the first year of the study shows that while 15 of the children with Down syndrome could say some of the number words and count objects, they tended to be using shorter count word sequences and to be counting smaller set sizes accurately when compared with typically developing children of the same non-verbal mental age. However, parental support improved the performance of both groups of children to the same extent and the same number of children in each group showed some understanding of cardinality by counting when asked to ‘give’ specific quantities.

The progress of preschool children with Down syndrome is likely to be affected by their ability to say the count words due to speech production difficulties and by the effect of their working memory difficulties on the learning of the number word sequence. Their counting ability may also be affected by their delayed fine motor skills when they are asked to count objects. The longitudinal data for these children will provide valuable information on rates of progress and the variability of achievement for this age group.

Studies of school age children

Two studies investigated the counting skills of school age children and came to different conclusions. In one study,^[1] junior-age school children as a group, were found to be not as able as mental-age matched non-disabled children in solving a novel counting problem and in standard counting and cardinality tasks, though there were two children with Down syndrome who were able to do the tasks. The group difference reported in this study could indicate that children with Down syndrome have a specific difficulty with number as they performed less well than would be expected for their mental-age.

In another study^[2] using the same types of tasks, but matching the children on a language measure, rather than a general mental age measure, the children with Down syndrome performed as well as the non-disabled children. This study indicates that the children are performing at the level expected for their language ability.

While these results may differ because of different matching measures, both indicate that children with Down syndrome progress through the stages of understanding number in the same way as other children, even if they do so at different rates relative to their abilities in other areas of development. Further studies are needed to explore more fully the links between number competence and other abilities for children with Down syndrome.

A study by Porter^[3] provided results consistent with the view that children with Down syndrome have particular difficulties with tasks utilising auditory sequential memory, compared with other children with a similar level of learning difficulties who were matched on vocabulary measures, all aged between 7 and 16 years. All of the pupils participated in a variety of counting tasks. The pupils with Down syndrome made few errors in one-to-one correspondence when counting items but the results showed they had relatively greater difficulties mastering the count word sequence. This finding supports the view that teaching practice for helping children with Down syndrome to count accurately should include rote counting and visual learning techniques (for example, number lines, number squares, materials that help children to memorise and ‘see’ the number sequence). Porter also considered that the additional demands of moving items and saying count words for items simultaneously (hand motor and speech motor) might make the task more difficult for pupils with Down syndrome than for other children, given that their fine motor skills may not be as good.

Evaluating interventions

Particular teaching methods and materials for groups of pupils with Down syndrome, for example, teaching addition using dot notation,^[29–31] teaching counting-on,[TODO: 6] money skills,[TODO: 16] conservation of length and quantity,[TODO: 14],[TODO: 15] have been evaluated for groups of pupils with Down syndrome. Sometimes these have a ‘control group’ of pupils who do not receive the particular teaching intervention. The wide variation in rates of individual progress is clear from these studies, some of which are single case studies or use only small groups of children.

Teaching counting and early calculation

Studies which have used dot notations and dice patterns, or dots on numbers as in ‘Touch math’, all report successes.^[29–31] These studies report highly structured teaching approaches, which make appropriate use of the pupils’ visual learning strengths, with tasks broken into small steps and with a great deal of practice built into the teaching programme. This structure and practise may explain much of the success of the methods rather than the particular notation system taught. The studies have usually only followed progress in the early stages of counting with small numbers to 5 and cannot provide evidence that the approach will help children to progress at later stages when they have to understand tens and units.

Therefore, the authors would be cautious about the use of visual systems such as dice patterns and ‘Touch math’, which do not support real understanding of the ordinal nature of the number system (that each next number is one more) or calculating with numbers greater than ten. The advantages of using a system such as Numicon [TODO: references 28] from the earliest stages is that it is a much more sophisticated system designed to give an accurate visual-spatial representation of the number system and to give children concepts and mental images to support their number development throughout their school years. However, Numicon also needs long term evaluation to confirm its potential benefits.

Teaching conservation of quantity and length

In two papers.^[32,33] Lister and colleagues demonstrate that children and teenagers with Down syndrome demonstrate wide variation in their understanding of the Piagetian concept of ‘conservation’ with respect to conservation of number, length, and volume. (Children understand conservation of number when they know that changing the arrangement of the items does not change the quantity. Before they understand this, if you put out two rows of 4 items in one-to-one correspondence and then spread one row to be longer, they will say there is more in the longer row. Similarly, understanding conservation of volume is achieved when children agree the amount is still the same if you pour liquid from a tall, thin beaker into a small, wide one. Until they understand conservation, they will think that they have less in the small, wide beaker).

The researchers divided the children (aged 8 to 19 years) with Down syndrome into two groups matched for their understanding of conservation and gave one group teaching to help them to understand conservation. The group that received the teaching soon showed significant gains in the number of children who understood conservation. The teaching methods were based on the original work of Piaget and Inhelder, and were essentially very simple demonstrations of equivalence. For number, different arrays of items were counted, and then put in one-to-one correspondence patterns to demonstrate equivalence. With liquids, they were poured back and forth between different shape containers to show equivalence. Language such as ‘’it goes back the same way’‘, ’nothing has been added or taken away’ was used by the teachers. The children’s verbatim language examples illustrated their understanding after training, e.g. ‘’Because it the same back together again’‘,’‘Because same juice pour it back’‘,’‘It will go back’‘,’‘Make it back in there’‘,’‘It is. This thing go in here and go back in there’‘,’‘It go in there it was’’.^{[lister_development_1989:p65?]} The children showed that they had retained their understanding when tested again two weeks later.

Teaching money

In our experience, many teenagers and adults with Down syndrome do not find money easy to understand but in order to fully understand the money system, children need to understand ‘place values’ (i.e. that 10 is the same as ten ones) and counting in tens, ‘fives’ and ‘twos’. Place value cannot be understood until children show a confident understanding of:

• cardinality (the last number represents the number of items in the set)
• the ordinal nature of the number system (each number is ‘one more’)
• the logical relationships between numbers (2 and 2, 3 and 1, are both equal to 4, or 6 and 4, 7 and 3, 8 and 2 all equal 10)

However, teachers often attempt to teach money to children who do not yet understand these basic facts about the number system. This view is confirmed by research in the USA published in 1996.^[34] A group of 17 children with Down syndrome aged from 10 to 18 years took part in tasks designed to assess their understanding of money. The children’s performance could be linked to their basic number knowledge and the researchers recommend working on these basic skills first, teaching counting by rote in ‘ones’, tens, ‘twos’ and ‘fives’ (depending on the country’s coin system - in the UK there are 1, 2, 5, 10, 20 and 50 pence coins, and £1 and £2 coins) and ensuring that children can count on. At this point card ‘coins’ are introduced with values written on them (1p, 2p, 5p, 10p, etc.) and the children taught to order them and then to understand their relative quantities (i.e. that 2 x 1p = 2p). The next step is to work with the children with comparisons of a) equal numbers of ‘coins’, but different values and b) equal value of ‘coins’, but different numbers of coins. The final step is to instruct the children in comparisons with different value and different number of ‘coins’, and then to move to real coins. The authors emphasise the importance of allowing children to master one step at a time and to master all the steps with card ‘coins’ clearly marked for value before moving to use real coins. The first step with the real coins is for the child to be able to state consistently the value of each coin when presented with them. The steps worked through with the paper coins are then repeated with the real coins. Again the authors emphasise the importance of mastering one step at a time with the real coins.

Many teachers point out the value of playing shops from primary school,^[35] for all children, and children with Down syndrome are certainly likely to be helped by using the card ‘coins’ in shopping games throughout their school years.

Single case studies

Single case studies, where pupils have been taught particular skills and their progress measured, continue to provide valuable information about teaching methods, rates of progress and individual learning.

Parents have described the progress of individual children using carefully planned programmes such as Kumon Maths [TODO: references 27] (a highly structured system originated in Japan), a Dutch dice notation system [TODO: 26], and the Numicon approach.[TODO: 28] All parents comment on the amount of practice that their children needed to master new skills and that they often had difficulty transferring the skill to use it in different situations.

While the few case studies available report the successful use of different approaches to teaching early number, there are some common features. The methods have broken down the tasks into small steps, with considerable practice at each step, to achieve results. It is not possible to draw any general conclusions from the case studies but they do serve to encourage higher expectations of achievement. The authors encourage teachers and parents to undertake and publish single case studies as they do add valuable detail to the knowledge base available. It is difficult for researchers to gather the detailed longitudinal information of children’s learning that parents and teachers can provide. If enough case studies are collected, then common elements leading to success or failure may become apparent and allow the information to be generalised to help other children.

Case study examples of high achieving secondary pupils

Dr Elizabetta Monari Martinez, a mathematician and researcher at the University of Padua, has documented the progress of pupils with Down syndrome included at secondary schools in Italy. Italy has had fully inclusive education for all pupils with learning disabilities for over 20 years. One of her publications^[23] describes two 15 year olds studying algebra with their peers [Figure 7]. Dr Martinez concludes that:

“Students with learning disabilities can succeed in academic programs, where even typical students may have difficulties, and can enjoy studying these programs. If we believe the academic culture is precious and pleasing for us, why should we not share it with people with difficulties? If it helps us, why should it not help them? I think the right path might be a fair balance between academic programs and training for autonomy.”

! Examples of work by Italian teenagers learning algebra

! Examples of work by Italian teenagers learning algebra

Figure 7. Examples of work by Italian teenagers learning algebra. [TODO: references 18] (Reproduced with permission)

The outcome of the study was that the two pupils, both of whom were typical of people with Down syndrome of their age and could be described as having mental abilities that fall within the category of severe learning difficulties, learned to calculate algebraic expressions, step by step, following the same path as their typical classmates, but at a slower rate, with more steps and with individual teaching. One was more careful and accurate than some of her non-disabled classmates.

When discussing the reasons contributing towards their good progress, Martinez comments on:^[23]

• The families of the students, who always trusted in their children’s possibilities in a realistic and consistent way and were always watching out, looking for the best.
• The strength and consistency of the students, who, as many persons with Down syndrome, can work a lot, if they are motivated, despite their difficulties.
• Inclusion in a regular class, which gives to the students the motivation to adapt themselves to their surroundings and to improve both their social and their academic skills.
• The professional ability of the special educator (special teacher).
• The collaboration between the teacher of the course and the special educator in adapting the mathematics curriculum and in teaching it.
• The choice of the mathematics curriculum and of its progression.

On the basis of her experience of teaching pupils with Down syndrome, Martinez argues for a revision in the way teachers often consider mathematics - especially for the slower learning child. She illustrates her view with the ‘old’ maths and ‘new’ maths trees ( [Figure 8] and [Figure 9]). The ‘old’ maths tree illustrates the view that a child cannot attempt any other branch of maths until they are competent at basic calculations. She argues that the ability to do arithmetic - especially mental arithmetic - is not essential to understanding aspects of other area of maths such as geometry, problem solving, measurement and data plotting.

! The ‘old maths tree’

Figure 8. The ‘old maths tree’.^[9] (Reproduced with permission)

! The ‘new maths tree’

Figure 9. The ‘new maths tree’.^[9] (Reproduced with permission)

She also argues that, as pupils with Down syndrome seem to often have difficulty with arithmetic, we should put ‘patches’ on for them by using concrete materials, visual prompts, and teach them to use rulers and a calculator.

The authors would support the view presented in the ‘old’ and ‘new’ maths trees, as in their own experience, pupils with Down syndrome have been held back from the wider maths curriculum in some schools, by teachers using the ‘old’ maths tree assumptions. However, when they have been allowed to try fractions, geometry, data plotting and measurement, they have enjoyed these topics, acquired some skills and understood at least their early stages and applications.

The ability to use a calculator will clearly be a valuable adult skill and at least one eminent teacher in the UK has argued for more than 20 years that ‘the definition of basic numeracy is the ability to use a four-function electronic calculator sensibly’,^[36] but more research is needed here into the ability to use a calculator. Pupils need to learn the calculator procedures and they need to learn how to check the answer or know if it is clearly wrong (much too big or small a number for example).

What can we learn from research with children with Down syndrome?

There is wide variation in achievements in number and maths skills, with some children with Down syndrome showing skills appropriate for their chronological age but others having considerable difficulties with basic counting and calculation. The reasons for this wide range of individual differences in achievements are not yet understood and the maths potential of students with Down syndrome still to be fully documented.

• Achievements for the whole group seem to be improving with better education and higher expectations
• Achievements are higher when children are educated in mainstream schools
• Numeracy skills usually lag behind literacy skills, but it is not clear why
• Numeracy skills seem to be at the same level as language comprehension skills, but this could be due to in part to numeracy tests that are presented verbally
• Children with Down syndrome master the early steps in counting in the same way as other children, but at a slower pace
• Their ability to learn the number word sequence seems to be delayed for mental-age and may be affected by speech production and auditory short-term memory difficulties. They may make more errors in counting tasks because of working memory difficulties.
• Structured teaching, with tasks broken down into small steps and practised sufficiently, improve progress and develop new skills
• Teaching approaches that use visual supports to teach number seem to help but as these are also structured methods, more research is needed to identify the most effective visual support methods
• Conservation can be understood by many children and teenagers with Down syndrome if explicitly taught
• Some children can learn algebra if taught appropriately
• Children should be able to try all aspects of the maths curriculum and not be held back because they have difficulty with number calculations

What are the implications of the research reviewed for teaching and learning?

Research studies do not account for all factors contributing to variation, for example, the pupils’ differing abilities, interests, family and educational experiences, but they do offer pointers that may help all pupils to learn when implemented in practice in the classroom and at home.

In addition, strategies should take account of what is known about the development of number skills in typically developing children and the specific profile of strengths and weaknesses that are usually associated with Down syndrome (see box).

General principles for teaching numeracy to children with Down syndrome

The same stages in learning as other children

Children with Down syndrome learn about number in the same way as other children. They should be therefore be taught in the same ways and join in all classroom maths activities with support, noting that they may need to learn in smaller steps, and with more practice to achieve mastery of each step (differentiation of the curriculum). They will also benefit from the use of a wide range of materials for counting activities, which are relevant to their daily lives, in order to generalise their skills and make them functional. (Readers will note that these principles will be good practice for teaching some 20-25% of the children in mainstream classes).

Wide range of progress, good teaching helps

Children with Down syndrome vary widely in their rates of progress in learning number and maths. There is evidence that the teaching they receive influences this as well as their learning abilities (Readers will note that these statements apply to all children). The range of abilities seen in children with Down syndrome overlap with the range of abilities seen in the typically developing population, so that some children with Down syndrome will not be slower than other children in mainstream classrooms.

Social learning, everyday experiences and games

Children with Down syndrome have strengths in social learning. They enjoy interacting with parents and peers, and will learn successfully if parents and teachers make use of games and daily opportunities to teach them about number. Daily opportunities include counting items into the basket at the supermarket, counting favourite teddy bears, counting favourite CDs, and so on. Games, where each player takes a turn can be a very effective and fun way to teach new skills. Being part of a group can take the ‘pressure to perform’ off children and teenagers with Down syndrome, which they may feel in one-to-one teaching situations, and the other players can model the correct responses for them. The element of competition can also be motivating.

Speech production difficulties

For children with Down syndrome of all ages, their speech and language difficulties need to be considered. Their learning ability should not be underestimated because they have difficulty saying words. Practice at the number words, using both numerals and the written words, will help them to learn to say them. Learning to count and to understand quantity should start in preschool years and can be demonstrated visually, using numeral cards and signs. Teaching should not be delayed because the child cannot say the words.

Language knowledge

Because children with Down syndrome learn to understand the language more slowly than other children, there is a risk that they are not introduced to the words and concepts that they need in order to understand maths, early enough. A vocabulary list of the maths and number words children need when they start full-time school should be available to parents and preschool staff and should be used as a guide for teaching the children. The concepts are not always more difficult than those for words they already understand but sometimes they have not had exposure to the words in contexts where they can learn what they mean. Learning the language for maths will continue throughout school years.

Motor skills

Motor skill delays may mean that manipulating small objects for counting activities are more difficult and this should be taken account of when choosing materials. Later, writing will be delayed and support with numeral cards, a Teaching Assistant to help record pupils’ work, work sheets that give choices for the answer and the use of the computer are ways to help.

Auditory processing and working memory

Auditory processing and working memory difficulties can be alleviated by the use of visual supports for learning. Visual supports for learning about numbers include written numerals, number squares, times tables and calendars. The visual images for the numerals support the learning of the spoken number names. These visual supports enable the children to learn by linking a specific visual image with the spoken word for a concept. There is evidence that this will help them to learn more effectively than only hearing the spoken word.^[37]

Visual and multi-sensory learning helps

Visual or multi-sensory learning strengths should be used to aid the understanding and use of the number system and at the present, the Numicon system is the most suitable as the materials provide a clear visual-spatial representation of the ordinal nature of the number system (that each ‘next’ number is one more), of the relationships between numbers to support the understanding of addition and subtraction, and illustrate tens and ‘units for children. The Numicon activities support an approach to teaching number through the recognition of patterns, through play with the materials and through activities to help children ’see’ the whole numbers without counting by developing mental images for them. In our view it is an approach that is ideally suited to the learning style of children with Down syndrome and their experience is that children really enjoy learning with the materials. The only difficulty for some young children may be their ability to handle the shapes. The Numicon approach was designed to help all children to enjoy and succeed in learning about number and is being adopted in many primary schools in the UK to support the maths curriculum.

In this way, children are being helped to understand and use the number system with the aid of materials that they can manipulate to carry out operations and they are developing specific visual images to support recall of specific numbers. In fact the imagery encouraged is both visual and multi-sensory as children learn to identify the shapes by feel (finding them in a ‘feely’ bag) as well as visually.

The importance of practice and automatization of skills

Children and teenagers with Down syndrome will benefit from sufficient practice to enable them to learn number skills - i.e. to store the knowledge effectively in long-term memory and to be able to retrieve it when needed. This practice need not be repetitive and boring. For example, mastering the count word sequence in order, first 1 to 10 and then to 20, can be done with many different counting games and songs, and counting of different sets of items. The number names for 1-20 needs to be learned by rote in English but beyond 20 the systematic naming system means they can be deduced. However, practice in counting to 100, over time, will be useful in aiding quick recall. Experts suggest that practice in order to learn means practice for up to twenty minutes a day, not the completion of two worksheets.^[20] Each step in the learning of number facts needs to be practised to mastery from the early stages to the steps in understanding money.

Examples of pupils’ teaching, learning and achievements in the classroom

Development for young children aged up to 5 years

From 2 to 5 years, young children with Down syndrome will typically be learning about number and mathematical words through play, song and life experience. Most would be ‘working with’ numbers from 1 to 10. Ideally, by 4 to 5 years, they would also have exposure to higher numbers through rote counting activities and use of numbers in daily life. Early learning activities, as for all children, would typically include reciting sequences of numbers, counting, and understanding of small quantities. Sign language can compensate for weaknesses in speech, therefore a child does not need to be able to speak to learn about numbers. Young children should have mathematical language used with them, especially number words, as they would if they did not have Down syndrome.

Most 3 to 5 year olds with Down syndrome are capable of practising counting with number lines and other materials and can learn or begin to learn the stable order of numbers, numeral recognition and one to one correspondence from games and counting exercises, and see how quantities are the same and different.

They may be beginning to link quantities with number words and numerals and may be helped to do this by using regular number arrangements or patterns. Through play, construction games, social learning and pre-school activities they will be increasing their vocabulary knowledge for number words, prepositions, categories for comparing and contrasting features, instructional language, measurement and time and money words.

Development in primary school, age 5 -11 years

As pupils with Down syndrome get older, so the range of variability in all skill areas increases and the differences between individuals becomes greater. Rates of progress may change for individuals depending upon their health, hearing, quality of education and family life as well as their cognitive abilities. Steady, slow progress is valuable and should encourage educators and families to keeping teaching.

In the infant years (ages 5 to 7) all pupils with Down syndrome will be working on developing number skills up to 20, with awareness of numbers beyond 20. At age 7 some pupils will still be working on numbers to 5 and some will be able to count by rote beyond 20, read numbers to 100 from a hundred square, be able to add and subtract to 10, order numbers to 20 and ‘count-on’. Many will be joining in with the numeracy activities in the classroom, counting to bigger numbers, or counting in two’s, fives and tens, first as a memory game and later as a series of mental addition. All will be continuing to learn new vocabulary for maths. They will begin to extend their knowledge of time, learning the days of the week and hour times linked to important events of the day. They will also be introduced to money, learning the names of the coins and playing shops.

In the junior years, 7 to 11, many pupils will know something about numbers to 100, counting in tens, ‘fives’, ‘twos’, addition to 20, subtraction, early multiplication and division and will have increased their vocabulary knowledge. Some pupils can add larger numbers using learned procedures, visual and mental strategies. All pupils will continue to learn about time, money and measurement in small steps.

Some children will understand arithmetical operations but others, who are using learned procedures may be confused by changes in style of presentation, materials or language used. The most ‘maths able’ pupils will be learning how to manipulate tens and units to add and subtract larger numbers, often with the support of specialised materials such as ‘Dienes’, ‘Numicon’, ‘Cuisenaire’ or an abacus. Many will learn through explicit routines, for example, for adding two numbers together, put the larger number in your head and count-on. Strategies, routines, visual aids and mnemonics will help children to manipulate and remember procedures and number facts, as well as ways of remembering how to read tables, graphs, grids or ways of presenting data. Some pupils may not have mastered number to 10, although their skills will be improving. Many children with Down syndrome enjoy mathematics, at whatever level they are working, and a small number of pupils are good at mathematics and function within the range of typically developing children of similar age.

Recording

The pupils’ general problems in recording information, for example in handwriting, writing numerals, drawing objects and diagrams, may present them with difficulties. Encourage the pupils to practice their skills to improve them, but also use tools to make the task easier and prevent failure at a task they understand but are not able to produce on paper. Make units of sticky paper, templates, numerals on sticky paper, numerals on card, lines of blocks on strips of card (cut up number steps), pieces of card cut to specification, practical maths objects and aids, and use a computer and printer.

Development in secondary school, age 11-16 years

The variation in pupil’s skills and interests will continue to diverge in this age range. Some pupils may make gains in basic skills that they were finding difficult in their primary years, because of their increased maturity, improved work habits, attention and motivation. Many pupils respond well to the change in approach in secondary school, with greater emphasis on life skills, combined with greater independence at home and daily opportunities to use their practical number, money and time skills. Focused numeracy teaching by a specialist teacher in numeracy may also produce significant gains. Pupils may be successfully taught with other pupils experiencing difficulties learning maths and number skills. Mathematics understanding and number skills will continue to improve, and we have observed considerable development for many pupils during adolescence.

Learning about time, money and measurement to varying skill levels, are life skills that will help pupils to achieve independence and are likely to be continuing goals for all secondary age pupils. Even though many teenagers and adults do not fully understand the money system to the level that enables them to work out change accurately, most can learn the relative value of notes and coins (which are ‘bigger’ or ‘smaller’ amounts) and can handle their own money to shop and pay their bills, with minimal support. In our experience, real progress with time and money are often observed in the teenage years, when young people need these skills ‘for real’ in their everyday lives. For example, the daughter of the second author showed little interest in learning about money in the classroom but made noticeable progress at 16 years when she began to sign for and collect her own money at the Post Office. As an adult, she manages her own money with minimal help, even though she cannot count out correct amounts. She withdraws her own money, pays her rent, saves for holidays and clothes, and takes care of her weekly spending - with minimum help and planning from a support worker. She needed real life motivation to learn new skills as she demonstrated a little later when, at the age of 23, she learned to use the telephone in record time so that she could phone her boyfriend (after years of no progress in formal teaching situations).

There is a message here about the importance of identifying the everyday relevance of classroom learning by linking it to real events in pupils’ everyday lives as often as possible. This may only be achieved by close co-operation between teachers and parents

In summary

More research is needed into the development of number and mathematical skills for individuals with Down syndrome. Consistently good teaching through the school years will be needed before the achievements of individuals with Down syndrome reflect their potential. Improvements in the teaching of number and inclusion in mainstream classrooms have resulted in higher levels of achievement for pupils with Down syndrome. This supports the view that pupils need a high standard of teaching, using a variety of techniques and approaches, and daily practice. Children and teenagers with Down syndrome also need social inclusion at school and in the community to learn skills and apply them in everyday life. For some individuals, progress will be slow, but skills will accumulate over the years to become useful for independent adult living. Increasing numbers of young people with Down syndrome are becoming functionally numerate, and this pattern is expected to continue with access to better teaching at school, higher expectations within the family and at school, and greater opportunities to use their skills independently in the community.

Acknowledgements

The authors would like to thank Tony Wing, Romey Tacon, Ruth Atkinson, Vikki Horner, Mike Fluck, Joanna Nye, and all the parents, teachers and children who have shared their knowledge and experience of number learning. However, views expressed in the text and any errors are solely our responsibility.

Terminology

The term ‘learning difficulty’ is used throughout this module as it is the term currently in common use in the United Kingdom. The terms ‘mental retardation’, ‘intellectual impairment’, and ‘developmental disability’ are equivalent terms, used in other parts of the world.

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