number-adolescent

Introduction

During primary school years, the rate of progress of individual children with Down syndrome varies quite widely, with some making faster progress on the mathematics curriculum than others. Some teenagers with Down syndrome will have some understanding of numbers to 100, counting-on, addition, subtraction, multiplication and division, when they start their secondary education, but many will still be learning these activities with numbers from 0 to 20.

In our view, the mathematics curriculum during the secondary school years should focus on giving teenagers a basic set of skills for use in their daily lives. It will include learning to tell the time, to measure and to weigh, to understand volume and shape and to understand the money system. The core activity, necessary for all these applications, is learning to understand the number system - how to count, to understand that numbers represent quantities, and how to calculate using numbers. Clearly a basic competence in understanding and using numbers to 100 will be important if a teenager with Down syndrome is to be able to count, measure and weigh, tell the time and use money for daily activities such as shopping, cooking and for work tasks.

Example of an 11-year-old boy’s achievements (he enjoys maths, has received weekly individual teaching at his mainstream primary school and practises his skills at home)

• Can count beyond 100, with numeral recognition and sequencing skills
• Can count in 2’s, 5’s and 10’s and multiplication facts e.g. 3 x 10 is ?
• Can add to numbers greater than 10 mentally without visual support e.g. 12 add 4 (fairly easily)
• Can subtract from numbers greater than 10 e.g. 14 take away 3 (fairly easily)
• Able to add 3 numbers together with some type of visual support (written numerals, Numicon shapes, rods)
• Knows that rearranging 3 numbers (above) will produce the same answer
• Can add and subtract in two figure columns
• Knows coin and paper money names and relative values
• Can add coin values together
• Knows costs of items of interest, regularly purchased (e.g. pop magazines, computer games, hire videos, music CDs, drinks, sweets)
• Knows how to tell the time usefully, although not completely
• Knows address, today, tomorrow, yesterday, vocabulary and dates for each, days of the week, months of the year, first, last, comparative (-er) and superlative forms (-est)

This module provides examples and practical ideas to help parents and teachers to teach young people with Down syndrome. The focus is on teaching a set of core skills and these activities should be relevant in both special schools and mainstream schools. While inclusion of individuals with Down syndrome in mainstream schools is increasingly becoming the norm in many countries during primary school years, at present the majority of teenagers will be in special schools or classes - even in the UK. There is very little research into the number development of teenagers with Down syndrome but at present, the evidence indicates that they often find understanding number more difficult than learning to read.

In the first section, information is included on the range of maths achievements of teenagers with Down syndrome from a recent survey conducted by the authors, and some examples of individual achievements. This information illustrates the wide variation in individual progress, with some teenagers showing an aptitude and interest in number and others only mastering a simple level of counting by the time they leave school. The teenagers who have more difficulty with understanding number may still have functional skills for telling the time and for using money, based on experience in daily use and the targeted teaching of practical strategies.

In our experience, teenagers with Down syndrome will be more motivated to learn if they can see the application of what they are learning in their daily lives and this seems to particularly be the case for number, time and money skills. It is therefore important for parents and teachers to work together, as parents will have the opportunity to give teenagers practice in using the skills in real situations. For this reason, the practical ideas in this module do not assume a knowledge of teaching mathematics and include a range of ideas that can be used to teach teenagers at home and in school. Ideally teachers and parents will be working in partnership, as parents can help teenagers practice and generalise their classroom learning in real situations at home such as tasks which require counting or weighing or the use of money. Parents can also help by playing number games with their teenagers, as these can be an enjoyable way of providing the practice needed to become competent.

The activities and strategies for supporting teenagers’ learning are based on what is known about the learning difficulties of teenagers with Down syndrome and what is known about how typically developing children learn number. Teenagers with Down syndrome will usually have significantly delayed language and therefore they may still not have all the basic vocabulary for number, size, colour, shape and quantity that will be used in the classroom when they start in secondary school. For this reason, a list of number vocabulary is included in the module. The teenagers will have been making slower developmental progress and this may have resulted in less opportunity to learn the basic number concepts during their primary school years. Teenagers have usually experienced delay in developing fine motor skills and this may have led to fewer opportunities to manipulate, sort and count small objects.

In the classroom, therefore, some teenagers with Down syndrome will still need help to learn basic number concepts and support for the practical activities of counting. They also have specific verbal short-term memory difficulties and therefore they will be helped by the use of visual supports for their learning whenever possible - using practical apparatus, number cards and number lines for example. None of these difficulties are unique to teenagers with Down syndrome and many teachers will be using the same strategies to support learning for other teenagers with speech and language delay, motor difficulties or memory difficulties.

See also:

• Number skills for individuals with Down syndrome - an overview

This module is intended to be used in conjunction with * Number skills for individuals with Down syndrome - an overview*, which provides the reader with a summary of relevant research on number development for individuals with Down syndrome, and the rationale for the practical approaches recommended.

Development of number skills for teenagers with Down syndrome: Examples of achievements

What skills might teenagers bring with them to secondary school?

A 13-year-old boy in a mainstream school

• Can work with numbers to 100, including subtraction and division on paper
• Knows multiplication tables to 7
• Can name coins and notes, count out simple amounts of money and with help give appropriate money in a shop. He needs help to check his change

At the end of their junior years (age 11 in the UK), some children with Down syndrome will know about numbers to 100, counting in ‘tens’, ‘fives’ and ‘twos’, addition to 20, subtraction to 10, early multiplication and sharing (or division). Most will not have perfected these skills and will make errors, depending on the activity and the situation. They will also have increased their mathematical vocabulary knowledge. Some children can add larger numbers using learned procedures, which may include visual and mental strategies. Some children will still be working with numbers to 10 and practising counting to 20. Almost all children will be able to count to 10: however, activities in this module are included for the teenagers (10-15%) who have not yet learned to count or to recognise numbers to 10.

What might they achieve?

A 15-year-old boy in a mainstream school

• Can recite and write numbers to 100 and count objects to this amount
• Can add large numbers 1000+ if written in columns, can add numbers to 100 using fingers to help with mental addition
• Can subtract numbers written down and use fingers to subtract mentally
• Knows 2,3,4,5 and 10 times tables and finds multiplication easier to do than other operations. He finds division difficult to do
• Can recognise and name coins and notes, understands the relative values of coins
• Can count out simple amounts of money (about 5 coins) but needs prompting to give appropriate money in a shop and relies on the shop assistant or someone else to check his change

During their secondary years, more teenagers will achieve adding and subtracting skills, counting-on, will know all the combinations of numbers that add to 10 (number bonds to 10), will be working with numbers to 20, and counting to 100. Some teenagers will be working with numbers to 50 or 100, adding and subtracting confidently, while others will be carrying out learned procedures but may be easily confused by changes in style of presentation, materials or language used. Some teenagers will know how to multiply and divide sums and problems on paper and may have good learned knowledge about times tables. Some teenagers will be learning how to break numbers into units and tens to add and subtract larger numbers, with the support of equipment, such as cubes, ‘Dienes’ TODO: references 1, ‘Cuisenaire’ TODO: references 2, ‘Numicon’ TODO: references 3 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 teenagers to understand problems, use procedures and number facts, and read tables, graphs and grids. Some teenagers in secondary schools may not have mastered number to 20 by the age of 16, although their skills will have gradually improved. Many teenagers with Down syndrome enjoy maths, whatever level they are working at.

This overview of progress is based on our experience of supporting teenagers in schools. Further information on the range of current achievements from a recent survey is provided in the next section. This information indicates that teenagers with Down syndrome do not find number easy to understand. The teenagers in mainstream education have higher achievements in number, indicating that teaching in an environment with higher expectations does influence progress, as there is no evidence that the groups of teenagers in the different school placements differed in ability when they entered school at 5 years of age. TODO: references 4 However, there is no reason to assume that the achievements recorded in this survey represents what is possible. With the better understanding of the cognitive profile and learning difficulties of children with Down syndrome gained in recent years and the introduction of materials such as Numicon and computer programmes to support learning from 5 years of age, the authors would predict that the next cohort of teenagers may well do better and show a greater understanding of number.

A 15-year-old boy attending a special school

• Can recite and write numbers to 100 and count objects beyond 20
• Can add and subtract low numbers, cannot multiply or divide
• Can recognise and name coins and notes, understands the relative values of coins, makes amounts from £1 to £5 with coins, provided he is reminded to start with the largest coin
• Can give appropriate money in a shop. He relies on help to check his change

Work in Italy, illustrated in the Number overview, has provided evidence that some teenagers with Down syndrome can learn algebra, so it is clear that there is much more to be learned about how to teach number to children with Down syndrome.

The current range of achievements

A survey carried out in 1999 4,5 offers some specific information about the number achievements of teenagers living in Hampshire, England, during their secondary school years (for more detail of the study, see the DSii Education overview. The group of 46 teenagers (ranging in age from 11 to 20 years) had widely varying abilities, as might be expected in a representative sample. The 28 teenagers attending special schools were older on average (mean age 16 years 4 months) than the 18 teenagers attending mainstream schools (mean age 14 years 8 months). Placement in mainstream or special school had been determined early in these teenagers’ lives (at 5 years of age) by the area in which they lived rather than on ability, as only one geographical area in the county was supporting inclusion for the majority of children with Down syndrome at that time.

The percentage of teenagers achieving skills in each type of school environment are shown in [Table 1] to [Table 9]. The skills of individual teenagers are also informative and examples are given in the side boxes.

Tables 1-9. Achievements of teenagers between 11 and 20 years in the Hampshire 1999 study TODO: references 4, TODO: references 5

###### Table 1.

###### Table 2.

###### Table 3.

###### Table 4.

###### Table 5.

###### Table 6.

###### Table 7.

###### Table 8.

###### Table 9.

The number and time skills of the teenagers in mainstream education were at a higher level than the teenagers attending special schools, but the practical money skills of the teenagers in special schools were better than those of the teenagers in mainstream schools.

An 18-year-old girl attending a special school

• Can recite and write numbers to 100 and count objects to this amount
• Can add and subtract single digit numbers mentally and numbers to 100 on paper
• Can multiply and divide on paper
• Can recognise and name coins and notes and understand their relative values, count out simple amounts of money and give small amounts of money in a shop appropriately. She relies on others to check her change

The higher achievements in number and time of the teenagers who have been included in mainstream classrooms throughout their school career probably reflects differences in the time spent on numeracy and in the levels of individual support available in mainstream compared to special education placements. The teenagers included in mainstream have been fully included in age-appropriate classrooms, with Teaching Assistants to provide individual support for learning as needed.

There is daily mathematics teaching in the UK primary school curriculum. In these mathematics lessons, the children with Down syndrome will have been working at their own pace on an individualised programme, in a classroom where all the children are engaged in numeracy activities. The comparison teenagers have been in special education classrooms where they do not usually have individual support and, even though numeracy is on the curriculum, it is difficult to teach with the same levels of intensity or expectation, given the diverse needs of the students in the special education group.

The better money skills of the teenagers in special education may partly reflect the difference in age and experience of the two groups, but is also likely to be linked with the emphasis of the special school curriculum on practical money use and life skills.

An 18-year-old girl at FE College (having attended mainstream school)

• Can recite, write numbers and count objects to 20
• Can add and subtract amounts below 20 in simple sums, can not multiply or divide
• Can recognise and name coins and notes, but does not understand the relative values of coins. She is not able to count out simple amounts of money but can add two 50p coins to make £1
• Can not give appropriate amounts generally in shops but always offers a £1 coin when buying sweets, as she knows this is enough. She relies on others to check her change

The reader should note that the teenagers in the survey range in age from 11 to 20 years and that these figures are means for the whole age group. The information collected in the study has been analysed to see if skills do improve with age, and to see if there is any difference in the progress of boys and girls (see [Table 10] and [Table 11]).

The teenagers were divided into three age groups: 11 years to 13 years 11 months, 14 years to 17 years 11 months, and 18 to 20 years. For money and time skills the data suggests steady progress, with older teenagers having better skills than younger ones. However, there was no evidence of progress with basic number skills (arithmetic). In fact the youngest age group tended to have the best scores in both school systems. This may reflect higher expectations and better teaching in primary schools for this group compared with the two older groups, as there has been a drive to raise standards in UK schools, including the introduction of a National Curriculum.

However, another possible explanation for no evidence of progress in arithmetic in secondary school years could be a lack of teaching focused on basic skills. In mainstream secondary schools, pupils with Down syndrome may continue to use some maths skills across the curriculum, but they may not receive the focused instruction and amount of practice that they need to improve their basic understanding and use of number. In addition, in all secondary school systems, there may be a tendency to feel that it is too late to expect much progress and that, if number concepts were not understood in primary school, maybe the focus should be on learning practical strategies for the use of time and money rather than expecting progress in basic understanding and number skills.

Table 10. Are skills increasing with age?

Mean Raw scores from relevant sections of the Sacks and Buckley Questionnaire (SBQ)

N = number of teenagers in each group.
Total N = 41, as the 5 least able students from special schools are not included in these analyses. |

Table 11. Are there any differences in progress of boys and girls?

Mean Raw scores from relevant sections of the Sacks and Buckley Questionnaire (SBQ).

No significant gender difference.
N = number of teenagers in each group.
Total N = 41, as the 5 least able students from special schools are not included in these analyses. |

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

• Social experiences and exposure to number
• Teaching methods
• use of practical materials to support understanding of number relationships
• importance of practice and rote learning of basics
• Knowing the language and concepts for maths
• The relevance of the skills to everyday life
• Reading ability
• Motor skills for counting and recording (writing numerals)
• Working memory capacity
• Logical reasoning ability

The authors believe, from their experience of supporting children in school, that all these explanations are plausible. They are aware that a number of the teenagers in the study did not improve their skills in secondary school but they might have done so if they had continued to receive some regular, specific instruction in basic number as part of their curriculum.

The older teenagers do have better time and money skills, mainly reflecting practice in everyday use. They are able to improve their ability to tell the time in hours and half-hours, and to give amounts of money within an appropriate range, by learning specific strategies. No one in the groups could divide a period of time into hours and minutes, or minutes and seconds. No one in the groups could count accurate amounts of money from any array of coins, or work out change. In order to be able to do these calculations with money, the teenager will need to be able to count to 100, understand ‘hundreds’, ‘tens’ and ‘units’, and ideally to count in ‘tens’, ‘fives’ and ‘twos’.

The point being stressed is that basic number skills are necessary in order to fully understand and use money, time and measurement.

Practical strategies, such as knowing which coins are needed to buy a soft drink, or for the bus fare, are very valuable in increasing independence and competence. However, we believe that many teenagers could increase their understanding of number and counting abilities, and that more time should be focused on this in the curriculum.

There were no significant differences between the boys and girls on any of the measures although, as might be expected from the data on typically developing children, the three most able teenagers in arithmetic were boys. The figures include achievements of teenagers with widely varying language and cognitive skills as well as number skills. However, achievements in language and other cognitive skills are not always consistent with performance in number for individuals. In our experience, interest, amount of practice and learning opportunities have a significant effect on progress with number.

In summary

The figures below are the combined percentages of special and mainstream data for the whole group:

Number

• 23% know numbers to 100, with the majority knowledgeable about numbers to 20
• 23% can write numbers to 100, with the majority able to write numbers to 20
• 25% of the teenagers can count objects beyond 20 with most able to count objects up to 20
• 81% can add small numbers
• 60% can subtract small numbers
• 17% can multiply small numbers
• 11% can divide small numbers
• 2% understand place value for 1000’s
• 5% understand place value for 100’s
• 33% understand place value for tens

Time

• 77% know the days of the week in order
• 67% know the current day if asked
• 43% know the month of the year if asked
• 68% understand a clock or timer
• 54% can tell the time by the hour
• 40% by the half hour
• 33% by the quarter hour
• 21% by 5 minute segments

Money

• 43% can identify some coins
• 18% can identify some notes
• 23% know relative values of some coins
• 41% can count some simple amounts of money
• 19% can give an approximate amount in shops
• 98% rely on shop assistants or others for change
• 43% have saved for a purchase
• 2% earn money
• 5% have experience of budgeting

The information obtained from this study indicates that some teenagers are only just beginning to learn to count and to understand number, therefore activities to teach the earliest stages of number and counting are included in the module. Teenagers who have been struggling with number may benefit from a new approach such as using the Numicon materials or computer programmes.

In addition, the large secondary school environment, more access in the community and shops in teenage years, and greater independence at home should provide many learning opportunities for teenagers with Down syndrome of secondary age.

Teaching teenagers with Down syndrome

Compared with other students, teenagers with Down syndrome need more teaching and practice in order to learn about number and maths. In order to plan effective teaching, parents and teachers need to take account of the specific language and cognitive profiles associated with Down syndrome (see ’ [A specific developmental profile]’ box). Teaching strategies can then be planned to make full use of the teenager’s visual learning strengths and to compensate for their auditory learning weaknesses. Activities, supports and teaching targets that will influence progress are listed below and will be developed further in the following sections of this module.

• Experience of numeracy - at home and in school through social interaction, saying numbers with others, counting, dice games, board games, wall displays, hearing number used during activities with everyday items and through teaching games
• Motor skills - handling objects, construction, speech for saying numbers, practice moving items, listening and speaking simultaneously
• Language for maths and number, including words for comparing, contrasting and categorising
• Drawing attention to quantities in play, sports, television programmes and daily routines, using fingers and other visual cues
• Learning to count, i.e. one word for one item when counting, knowing the order of numbers, understanding that the last count word represents the whole number (cardinality)
• Using a number line, hundred square and written numerals, for visual support
• Using practical materials which represent the number system visually to support their learning
• Learning to recognise patterns, matching patterns and arranging items into patterns
• Learning to recognise numerals, reading from a number line and a 100 square
• Using money, in practice and especially in real situations
• Using daily and weekly calendars to develop understanding of time
• Using a watch and clocks to understand time and plan ahead

Mathematics has a strong visual element and this can often be used to illuminate meaning. Visual teaching methods include frequent use of a number line, a 100 square, number apparatus, pictures, diagrams, graphs and computer programs. Teenagers can often be included in maths lessons by teaching ideas that can be shown visually, for example, geometry, ratios, fractions, data handling and algebra. Games and puzzles, where the rules can be picked up quickly by watching a demonstration, will also help teenagers with Down syndrome to learn and understand mathematics. Structured teaching methods and supports for language and memory are described later in the module.

A specific developmental profile

Teenagers with Down syndrome are helped by teaching methods which take account of research into their strengths and weaknesses:

• Their motor skill delays, making manipulating small items, drawing and writing difficult
• Their speech and language delays, leading to their understanding being underestimated
• Their auditory processing and working memory difficulties, making learning from listening difficult
• Their strengths in social understanding and enjoyment in learning from social interaction with peers and adults
• Their relative strengths in visual processing and visual memory, making learning from seeing important and effective; they are visual learners
• Their strengths in using gestures to communicate and in showing their understanding by pointing to or choosing an answer

For a full discussion of these issues, see * An overview of the development of teenagers with Down syndrome (11-16 years)*

Understanding number and mathematics

Primary skills list

The lists below for number, money, time, other measurements, shape, place and data, summarise the skills and concepts that teenagers with Down syndrome are likely to continue to learn about in secondary school. These are followed by an advanced skills list. The items in each list are not strictly in the order in which they will be learned. This will vary for individuals - some of the targets will be learned gradually over many years. The authors have selected some skills (in bold type) as particularly important achievements, as they each represent a significant step forward in understanding and using number.

Number

• Reading numerals 1 to 10
• Reliable counting to 10
• Counting principles to cardinality - ‘how many?’
• More, less
• Reliable counting to 20
• Order amounts
• One more, one less
• Bigger, smaller
• Recognise and understand 0
• Read, write and order numbers 0 to 20
• Count backwards to 0
• Count on
• Count back
• Use ruler
• Use calculator
• Begin to add and take away
• Recognise and name signs for +, - , =
• Procedures for adding and subtracting
• Know doubles (2+2, 4+4, 5+5, 1+1, 3+3)
• Choose the larger number of two and count on from the larger number
• Place value, units, tens, hundreds
• Count, read, (write) and order numbers to 100
• Know about odd and even numbers
• Know subtraction is the reverse of addition
• Know number facts for addition and subtraction to 10
• Know multiplication and division symbols
• Times tables, 2’s, 10’s, 5’s
• Recognises and can say number to 1000 (for scores, buses, cooking, competitions)
• Column addition and subtraction
• Understand division is the reverse of multiplication

Money

• Know coin names
• Understand coin values
• Add coins with values to 20p, 50p and then larger amounts
• £ and p notations
• Use money (with or without full understanding)
• Understand decimal notation for representing values in money

Time

• Days of the week in order
• Read hours on clock
• Seasons of the year
• Know the months of the year
• Read hours and half hours
• Read time to quarter of an hour
• Use units of time - seconds, minutes, hours, days, weeks, months, years
• Tell the time
• Use time to plan ahead, e.g. in cooking, for T.V. viewing, for special events

Other measurement

• Use ruler and scales for measurement
• Compare lengths
• Measure, weigh and compare

Shape, place and data

• First shape names and concepts
• Position, direction and movement vocabulary and concepts (see vocabulary lists on p. 47)
• Understand simple graphs
• More advanced shapes
• Identify right angles
• Identify symmetry
• Make simple tables and graphs of data
• Recognise fractions
• Use fractions

Completing this list is a great achievement

At this stage, simple word problems can be converted to number problems, number problems can be worked out, time and money are understood and can be used, measures can be used and compared, calculators and rulers can support mental strategies, and time across the year can be understood and used.

We know that many teenagers and young people with Down syndrome find converting word problems to number problems, telling the time, understanding time across the year and calculating mentally for using money difficult to do as quickly as other teenagers at this stage, or as quickly as they need to in real situations. But young people with Down syndrome can learn and use these skills, given extra time, the use of paper, pencil and other supports, and positive emotional support to encourage them to persist. Some teenagers with Down syndrome achieve more than this, and with their peers will learn additional skills, described below as ‘advanced’ for teenagers and young people who have Down syndrome.

Advanced skills list

Number

• Symbols for <, >
• Rounding up and down
• Mentally add or subtract two digit numbers
• ‘Times’ tables to 12 by rote
• Remainders after division
• Mental strategies and paper and pencil to solve word problems
• Work with bigger numbers to 10,000
• Long multiplication
• Use multiplication, division, addition, subtraction to solve simple word problems with a series of operations
• Decimals, percentage
• Ratios
• Algebra

Shape and data

• Formula and measures for area
• Parallel and perpendicular lines
• Measure angles with protractor
• Perimeter and areas of shapes
• Plot co-ordinates for negative and positive quadrants
• Interpret and extract information from tables, graphs and charts to solve problems

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Figure 1. Numicon shapes and pegs

Learning fundamental number skills

The basic ideas for learning about number come from noticing visual patterns, from learning about the order of the number system (how this always stays the same) and from counting experiences. Counting teaches children about number words as labels, the order of the number system and how to use numbers to find out how many there are. Learning to count will not necessarily have taught teenagers to understand the nature of the number system, and the use of materials which provide an accurate visual spatial representation of the system (such as Numicon and Cuisenaire) may help them to do this more fully. TODO: references 6 Numicon (see [Figure 1]) has been demonstrated to improve the maths progress of typically developing primary school children in the school where the materials and activities were developed. TODO: references 6 It is being used successfully in secondary schools, as well as primary, to support the teaching of children with Down syndrome and other children with numeracy delays. TODO: references 14

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 in current teaching practice 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 to 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. TODO: references 7

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.

Visual learning

See also:

• An overview of the development of teenagers with Down syndrome
• Number skills for individuals with Down syndrome - An overview

The visual memory and visual learning strengths of young people with Down syndrome can be used to support their learning of all aspects of the number system. Quantities or amounts can be seen, practised and memorised as a whole (e.g. that is ‘3’ items, that is ‘4’), as well as being units ‘to count’.

Teenagers will be helped to visualise or see number patterns and whole numbers by using a visual representation of the number, for example Numicon shapes, teaching materials and activities ( [Figure 1]). Numicon materials are available for a single child or for whole class use. TODO: references 3 The Numicon materials illustrate the number system by using a set of shapes designed to clearly show that each ‘next’ number is one more. In addition the shapes can be fitted together to illustrate addition and subtraction. Units or ‘pegs’ are provided for counting and pattern activities. These pegs fit into the holes and the shapes. Learning the patterns and values of Numicon shapes will provide a strong foundation for learning later number skills, like addition, subtraction, number bonds to 10, strategies for mental arithmetic and place value.

Introducing Numicon to teenagers who have found number difficult may provide a fresh start and help to make number more interesting and easier to understand.

Counting practice

The skills and understanding needed for successful counting have been defined as the one-to-one principle, the stable order principle, the cardinal principle, the abstraction principle and the order irrelevance principle TODO: references 7 ( [Figure 2]). These principles can be learned through structured games, including games with whole numbers, recognising patterns and other types of visual imagery. There are no short cuts to understanding number - each of these principles needs to be understood.

Many teenagers with Down syndrome in the 11-16 year age range have mastered these skills and achieved cardinality, or an understanding of ‘how many?’ The following activities will be useful for teaching teenagers who have not mastered counting to 10 or do not yet understand cardinality (see c in [Figure 2]).

1. The one-to-one counting principle. The child must use one and only one number word for each item to be counted, and not skip any item or double count any item

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2. The stable order principle. The child has to know the number words in the correct order and always use them in the correct order when counting

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3. The cardinal principle. The child understands that the last ‘count word’ represents the number of items in the counted set. At this stage, the child can answer “How many are there?” questions correctly and can give small sets of items correctly in response to “Give me … (2, 3 or 4) …” questions.

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4. The order-irrelevance principle. The child understands that the order in which items are counted is irrelevant

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5. The abstraction principle. The child now understands that any items can be counted (i.e. that quantity is a concept which can be applied to any type of items). Once they realise that the spatial arrangement of the items is also irrelevant they are said to understand “conservation of number” - a significant Piagetian step in cognitive development.

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Figure 2. The ‘how to count’ principles - and steps in understanding number, based on Gelman and Galistel. TODO: references 8

Learning number words in order

• Matching numeral cards, learning to select them by name and name them ( [Figure 3])
• Pointing to numbers on a number line to 10 and saying the number ( [Figure 4])
• Matching numeral cards, to their position on a number line ( [Figure 5])

Counting with a number line will help to establish the order of numbers and help teenagers learn to say number words more clearly through practice. Teenagers should first use the number line to learn the sequence of numbers to 10, and then to 20.

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Figure 3. Matching numeral cards

! Counting with a number line

Figure 4. Counting with a number line

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Figure 5. Matching cards to a number line

Learning about quantity

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Figure 6. Matching a numeral to a Numicon shape

Understanding quantity and the labels applied to differing amounts requires considerable practice, and matching games or games with prompts or visual cues for quantity will help teenagers to learn this skill. Numicon activities can help, as the shapes indicate the quantity represented by each number.

Initially teenagers will learn to:

• Match Numicon shapes (to 5) and then 10
• Match numeral to a shape ( [Figure 6])
• Select shapes by name
• Match shape to a number line
• Order shapes ( [Figure 7])
• Match shapes to appropriate quantities of pegs or other items

Matching quantities to numerals

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Figure 7. Ordering shapes

Teenagers can also be taught about quantities by using errorless learning methods, by being offered only the correct amount of items to match to the numeral (or shape). For example, a person may be asked to put two and three items into containers (labelled with the numerals 2 and 3) with 2 and 3 items placed near each container.

Teenagers will need help to understand the abstract nature of numbers - for example that groups of the same number but different types of objects are all sets of ‘3’. Explain this to them by showing them several groups of 3 objects, counting each set and placing a numeral ‘3’ with each set. Do the same with other numbers, first 1 to 5, then 6 to 10.

‘Giving’ the whole set

To build their understanding of cardinality, teenagers can be asked to give the whole amount of items that they have (with numeral shown) for small sets of 2, 3 or 4 items.

! ‘Giving’ 3 items from a larger set

Figure 8. ‘Giving’ 3 items from a larger set

‘Giving’ a number of items from a larger set - leaving some behind

Teenagers also need to understand that, when asked for a number of items from a group, this does not mean that they should count or give all of the objects. It means that they should give some and leave the rest ( [Figure 8]).

Games to practise counting part of a set and leaving some uncounted will help to teach this. Teenagers should be supported in these games at first, so that they do not make errors, and get used to leaving some items behind. It is our view that teenagers may not receive enough modelling or practice in this type of activity. Counting games usually require the teenager to count ‘all’ in a group and they then find it hard to stop counting part way, in order to ‘give’ a smaller set from a larger one.

‘How many’ covering and remembering games

When teenagers have practised matching the correct amounts, practise remembering ‘how many’ there are, by telling them how many items there are, for example ‘’1,2 (as you count) - there are 2 (pens)’‘. Then cover the objects (or pictures of objects). Make it fun by saying’‘How many (pens) am I hiding?’’ If they do not answer correctly, reveal the pictures or objects and say ‘’Look, there are two (pens)’‘. When the person is successful at this task, let him or her count the items before covering them up. This task can be continued, gradually adding variations, so the teenager is helped to understand that counting tells us ’how many’ of something there is.

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Figure 9. Using number stickers to record the answer

Rearranging the same set, ‘guessing’ and counting again

Games where the objects are counted, a numeral presented and then the same objects rearranged, followed by asking the teenager how many there are now, will help to develop a more conceptual understanding of number (conservation of number). Repeated counting of a set of items laid in different arrangements in this way, with discussion with an adult, will allow the teenager to realise that no matter what arrangement they are in, four items are still four items.

Learning to write numerals, number words and to use worksheets

Activities for practising basic number skills are often presented on worksheets or work books in the classroom. Developing confidence with paper and pen activities can help teenagers to work independently in a group setting. Teenagers will be helped by becoming familiar with the different ways that work sheets may present work and how they should respond to them. The responses typically required include circling numerals, number words or items, colouring them or drawing lines between them to associate or pair items or sets together.

Teenagers will also be able to practise learning how to write numerals and words through paper and pen activities. Those not yet able to write numerals can also work with number stickers ( [Figure 9]), number cards or plastic/magnetic numbers for written numerals to demonstrate their understanding and skill with numbers. Developing reading, writing and worksheet skills enables teenagers to work more independently in the classroom.

Learning about ‘one more’ and ‘one less’

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Figure 10. Examples of word and symbol cards, Numicon apparatus and ‘steps’

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Figure 11. Knowing the pattern of numbers to 20 backwards, forwards and in 2’s

When teenagers can count and understand quantity to 10, they will be helped to move ‘up and down’ the number system by practising ‘one more’ and ‘one less’ through structured teaching activities. Teenagers with Down syndrome are likely to need more practice to understand these concepts and how they can be used at any place in the number system. The language for ‘one more’ and ‘one less’ will have been used with them in their counting activities, but some extra practice is recommended, using number steps and other visual apparatus such as Numicon shapes, so that they can see how ‘one more’ means go up one, and ‘one less’ means go down one. Flashcards with ‘one more’ and ‘one less’ written on them can be an effective aid (see [Figure 10]). Practice sums that use this language, interchanging ‘one more’ and ‘+ 1’. Teenagers will be helped to use their skills by knowing the pattern of the number system forwards and backwards ( [Figure 11]).

Learning about bigger numbers

Teenagers with Down syndrome need a firm foundation on which to build their knowledge about bigger numbers by mastering numbers to 10. However, while achieving this, they also need to hear the words for bigger numbers to 20 and beyond, so that they can discriminate them from the lower numbers they are working with. They will need practice to help them recognise the new number words they hear, to say them and to associate them with numerals and written words ( [Figure 12]).

Using written words may help some teenagers to discriminate and remember new words, for example, distinguishing ‘fifteen’ from ‘fifty’. The numerals and written words can also be matched to their position on a number line, and this will be especially helpful for learning the ‘-ty’ words and ‘teen’ words.

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Figure 12. A visual support for practising saying ‘13’ and remembering its place

For learning to say numbers and learning the order of numbers for use in counting, teenagers should receive extra practice with all parts of the number system that they are learning about. Otherwise the numbers lower down the number system tend to be practised to the exclusion of bigger numbers. This can be achieved through continuing a count sequence over a period of days, or starting a count from a number anywhere on a number square, chosen by the teenager.

In the classroom a ‘spinner’ game or ‘roll the dice’ game can make choosing the beginning number more fun. Games with balls (e.g. counting the throws, turns or bounces), at home or at school, are particularly good for practising saying parts of the number sequence from higher up the number system. Alternate counting with a partner is another way to practise counting.

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Figure 13. A game to practise the steps for early addition

Learning to add

Addition with objects and fingers

When teenagers have mastered the counting principles for low numbers (to 10) they are likely to have begun to join groups of objects together to find out ‘how many?’ They should know the meaning of ‘more’ and that ‘more’ means adding. Teenagers can learn about adding and ‘add’ by joining groups of objects together to find out how many the new set makes.

Teenagers who have used teaching apparatus that represents number relationships, such as Numicon or Cuisenaire, will be able to see how two amounts join to make a new larger whole amount. They can also see how whole numbers can be broken into smaller parts.

Teenagers may use a variety of strategies for addition, depending on their rate of progress, but some will still be using the strategy of ‘counting all’ rather than ‘counting-on’:

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Figure 14. Symbol cards for a matching game

• At the ‘counting all’ stage, the teenager will need to use objects or fingers to add and, if asked to add 3 + 4 for example, will count 3 objects, count 4 objects, put them all together and then count the whole set, starting to count from 1.
• Once the teenager can ‘count-on’, they would be able to start their addition by counting on from 3 (e.g. 3, 4, 5, 6, 7) and not need to start counting from 1 (see [Figure 17]).

A framework or game, with space for teenagers to place numerals and objects or counters, can help them practise the steps for addition ( [Figure 13]).

Discriminating and learning symbols

The symbols for addition, subtraction, multiplication, division and equals can be learned on cards through matching games ( [Figure 14]), with the words written on the reverse ( [Figure 15]). Symbols on worksheets can be highlighted or emphasised in other ways while teenagers are learning them.

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Figure 15. Symbol cards with words on the back

Commutativity

Commutativity means understanding that 6 add 4 is the same as 4 add 6, or 3 add 2 is the same as 2 add 3. This skill will help teenagers to learn number bonds and enable them to speed up, automatise and reduce the errors they may make when working with numbers. Apparatus such as the Numicon shapes can help teenagers to ‘see’ how commutativity works (see [Figure 16]).

Doubles

Learning about doubles for numbers to 10 is a useful skill for adding (and subtracting) that will be used over and over again as teenagers move up the number system and work with larger numbers. Learning doubles can be a fun activity for teenagers to learn as an automatic skill, showing their fingers or just saying the answer. 1+1, 2+2, 3+3, 4+4 and 5+5 can be learned before teaching doubles beyond 5. Teenagers should also practise seeing how the identical sets combine (and split into equal parts), and they should complete written addition sums to practise their doubling skills.

Number bonds for addition to 10

Learning number bonds for addition to 10 (all the combinations of numbers that add to 10) will help the understanding of number and speed up arithmetic across the number system. These can be learned by rote as well as by practice through adding objects. Visual apparatus like Numicon shapes are especially helpful for teenagers learning number bonds - they can remember the arrangements they have practised and will know, for example, that a 5 shape and a 3 shape make an 8 shape, and an 8 shape and a 2 shape make a 10 shape ( [Figure 16]).

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Figure 16. Using Numicon to illustrate number bonds to 10

Counting-on

Most teenagers can learn to count-on from one number, continuing their count sequence with the second group, by beginning with the support of a number line ( [Figure 17]).

Counting-on example

The method Irwin TODO: references 9 used to teach counting-on to children with Down syndrome was extremely successful. The children she selected for tuition could count to 9, read and write numerals and demonstrate adding, but they always returned to number one to count groups of items together.

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Figure 17. Learning to count-on with a number line

Her teaching materials included a set of white cards with 6, 7, 8 or 9 black dots on them (long dot cards); a set of cards with 2, 3, 4 or 5 similar dots on them (short dot cards); and a set of cards with numerals on them.

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Three sub-skills were then taught:

1. The teenagers were asked to count aloud starting from a number greater than one, with as much help as was needed.
2. They were asked to give the cardinal name of the last dot of the first set, and shown that the numeral presented with the dot card (7 in the example illustrated) demonstrated both the cardinal value and the counting name of the last dot of the first set.
3. They were asked to give the counting name of the first dot of the second set (8 in the example illustrated), which required them to move from cardinal meaning to the count meaning of the number 7. Steps 2 and 3 were done in conjunction with one set of Irwin’s teaching cards.

Teaching prompts included:

• saying the appropriate number names with the children, mouthing initial sounds and letting the teenagers continue.
• taking them back to smaller numerals in their counting when necessary and gradually increasing the level of difficulty.
• close monitoring of the children’s activity with errors being corrected, allowing ample time for self-correction and giving descriptive praise for careful work and self-correction (i.e. praise that emphasises the correct steps the child had taken).

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###### Figure 18. Skilled addition, using number facts, counting-on and fingers/objects to assist

Teaching lasted for five days of one week, using cards and numerals on four days and blocks and numerals on the fifth day. All of the children learned to ‘count-on’ when adding, many of them on the first of the five teaching days. Children who were successful in their use of counting-on with a disordered array of blocks usually adopted the technique of using a printed numeral to help them remember how many blocks were in the first group. Six months later, all except one of the nine children continued to count-on with the specially made teaching materials, most used the technique when adding a random array of blocks, three children used counting-on when doing written sums and one when adding money.

This successful strategy applied important teaching and learning principles which can be used in any learning situation. These include analysing and breaking down the learning required into small steps, and the use of visual aids to help the children remember and learn those skills. When counting-on has been learned, adding can become faster, and combinations of mental strategies, fingers and items can be used ( [Figure 18]).

Starting with the largest number

Another skill that helps counting-on and adding is choosing the largest number to start with, so that it is easier to count-on. This requires a ‘sorting’ step before beginning the addition. Practice at re-writing or ordering the numbers in written sums so that the larger number comes first may help this, as well as to reinforce the idea that the answer is the same whichever way the numbers to be added are written. With apparatus for whole numbers, it is also easier to find the largest piece of apparatus first.

Teenagers can learn for some types of sums to put the largest number in their ‘head’ and to count-on if the number to be added is quite small (below 10).

With numbers above 10, teenagers can split the tens from the units to add them, so that 15 + 12 becomes 10 + 10 + 5 + 2. Apparatus that depicts tens and units visually is likely to make this task easier to do, especially as teenagers can see when the units make a ten.

Learning number facts

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Figure 19. Number squares and tens cards to help teenagers see how tens work in the number system

Number facts (e.g. number bonds, doubles, counting in 2’s, 5’s, 10’s) are essential for mental arithmetic of this type, including facts for numbers below 10 and for larger numbers (tens) and later, 100’s.

Using a 100 square

Teenagers will gain from practice of working with tens to 100 so they become fluent with the sequence of tens (10, 20, 30, 40, 50, etc). The regular pattern of the numbers, counting in tens, seeing how tens fit together to make 100 with 10’s cards and learning the ‘ten times’ table will help them. They can practice adding 10 to any number in a number square, such as 12, 22, 32, 42. They should understand how to find ‘10 more’ for any point in the number square (and how to find ‘10 less’) ( [Figure 19]).

Writing down points of ‘mental’ calculation to support memory

Mental arithmetic requires a good memory, especially when numbers are being split into tens and units for addition. Teenagers can use a pen and pencil to help them add ‘mentally’ so they do not forget the parts they have separated, for example, for 25 + 23, they can write down 20, 20, 5, 3 and then add these together. This is a combination of mental and written arithmetic, and teenagers with Down syndrome will need to record the steps to support their memory at each step in the calculation.

Using a calculator for addition

Teenagers could also do the separating mentally, write the numbers down and then add the numbers using a calculator.

Learning how to separate numbers into tens and units will help teenagers to check an answer they have found for a sum using a calculator and will provide practice in understanding place value.

Column addition

Column addition is useful for finding out answers for 2 and 3 digit numbers. Squared paper and sufficient gaps between sums will help teenagers learning this skill. Many teenagers with Down syndrome enjoy following specific procedures that they have learned in order to complete column addition. However, teenagers should be encouraged to check each of their answers using other strategies, as errors can be made easily by placing a number in the wrong column. With support, understanding of place value can be learned by completing column addition, combined with other teaching methods.

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Figure 20. Word and number cards to aid discrimination and memory

Learning about place value

Understanding place value means understanding the value of a symbol in a number system, dependent upon its position, i.e. understanding the notation for hundreds, tens and units.

First, teenagers need to be familiar with bigger numbers. After learning with numbers 1 to 10, teenagers should learn to read and name numbers in order to 20, and then to see how tens fit onto a 100 square ( [Figure 19]).

! Circling activity to assess discrimination of spoken ‘ten’ and ‘teen’ numbers

Figure 21. Circling activity to assess discrimination of spoken ‘ten’ and ‘teen’ numbers

In English, the numbers from 11 to 19 are spoken either in a unique way (11, 12, 13) or reversed from the way they are written (e.g. 14 = four-teen), which many children find confusing. Therefore the pattern of the numerals is likely to be easier to learn about than the words we say for them.

This is another reason why working with numerals in tens lines and 100 squares is beneficial - trying to understand values from the spoken words is often difficult for people with Down syndrome.

Teen numbers and the numbers ‘twenty’, ‘thirty’, ‘forty’, ‘fifty’, ‘sixty’, ‘seventy’, ‘eighty’ and ‘ninety’ should be read, said, listened to and related to their number position often, so that teenagers discriminate them and can say them. They need to know about the different types of words to distinguish ‘forty’ and ‘fourteen’ when they are listening - if they only know about ‘fourteen’ they will perceive ‘forty’ as 14 when they hear it said. Duplicate word cards for the -ty words and teen words used in matching and sorting activities help to teach differences, so they can be seen as well as heard and spoken ( [Figure 20]).

! Columns for learning place value

Figure 22. Columns for learning place value

Given the high incidence of hearing and auditory discrimination problems, it is important to check each teenager’s ability to discriminate spoken ‘-ty’ and ‘-teen’ words (see [Figure 21] for an example).

Teenagers will learn about the ‘tens’ and ‘units’ positions and later about ‘hundreds’, ‘tens’ and ‘units’, and how the value of the number relates to its position or place in the written number. When they begin to work with numbers on paper, the headings for the columns should always be written down for them. Large columns on a large piece of paper will make this easier for early place value activities ( [Figure 22]). Teenagers can then practice matching the numerals in the number to their position or place.

Numicon activities can help teenagers to understand, and to show to others that they understand place value. Written numbers above 10 can be shown with ten shapes and unit shapes, numbers can be spoken or read for teenagers to find the shapes and to find the correct numeral symbol. The activities provided with the equipment teach place value. If the activities have been taught in order from the beginning, teenagers will know that 12 is ‘ten’ and ‘two’ and that 33 is ‘thirty’ and ‘three’.

! Numicon shapes showing 12 and 33

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Figure 23. Place value cards TODO: references 10

For teenagers who cannot write numerals easily, numeral cards can show the whole number for the teenager to copy.

Place value cards with a triangle or arrow next to the unit can help teenagers to understand place value ( [Figure 23]). TODO: references 10 The cards are placed with the ‘unit’ card on top of the ‘ten’ card to illustrate how larger numbers are made up of ‘tens’ and ‘units’. In a similar manner, ‘hundreds’, ‘tens’ and ‘units’ cards can be stacked on top of each other, so that the difference in ‘value’ according to ‘place’ can be seen. The teenager learns that the triangles must always be on top of each other to ensure the cards are placed correctly.

Games of exchange

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Figure 24. A game of exchange: counting 12 in tens and units blocks

Games of exchange, using various types of apparatus, help teenagers to learn about hundreds, tens and units and how these relate to each other. Games designed for this purpose are available in most (primary) school resources. Colour changes or shape changes can be used to show hundreds, tens and units too, so that when ten red ‘unit’ blocks have been counted they are exchanged, for example, for one white ‘ten’ block, and so on ( [Figure 24]).

Clear and well presented visual games, flashcards and other types of teaching and memory aids are included in Count Us In! a pack to support the UK Numeracy Hour Curriculum for primary schools (Key stages 1 and 2). TODO: references 11 Most of these activities are suitable for teenagers with Down syndrome working at these levels.

Subtraction

The idea of absence, removal, taking away, ‘one less’ and ‘nothing’ are less common in everyday life than ‘adding’ and teenagers almost always find subtraction more difficult than addition. It is likely that teenagers with Down syndrome receive far less practice in subtracting than adding. Many children who do not have learning disabilities can adapt the strategies they have learned for addition to subtraction, so at the stage when they are learning about subtraction they may be relying less on objects and apparatus and more on mental abilities than when they learned about addition. It is therefore important to offer teenagers with Down syndrome the same amount of practice that they had with addition if they are to understand subtraction. Teenagers should be encouraged to look carefully at the symbol, or have a clearer than usual symbol, to be sure whether they are adding or subtracting.

Understanding how addition and subtraction are linked with each other will develop through practice. A game with a spinner or dice showing numbers to be taken away (e.g. -1) or added (+2) used with a number square is an enjoyable way to practise ( [Figure 25]).

Teenagers should also practise reversing addition and subtraction with, for example, written sums, checking their answer by adding numbers back together, breaking up numbers and joining them together (using apparatus), to help them understand the relationship between adding and subtracting.

Just as teenagers needed to know that ‘more’ usually means ‘add’ or ‘give’ so they need to know that ‘less’ usually means ‘subtract’ or ‘take away’.

Strategies for subtracting include:

! Subtraction and addition board game

Figure 25. Subtraction and addition board game

• Counting out the larger number using items or fingers, counting and then removing the smaller number and then counting what is left
• Counting backwards from the larger number to the smaller number, keeping a tally on fingers
• Counting up from the smaller number to the bigger, keeping a tally on fingers
• Counting from the smaller number to the bigger, using a number line
• Using a number fact and reversing it, for example, 4 + 4 = 8, so 8 - 4 = 4 (these can be learned by using Numicon shapes and activities)
• Using a number line to count backwards from the large number, so that the backwards counting sequence and the start and finish points are not forgotten

Learning the steps for subtraction with apparatus can help teenagers to understand subtraction. A framework can help teenagers to remember the way to complete a sum, help them to keep track of what they have done, and to know what they need to do next ( [Figure 26]).

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Figure 26. Subtraction framework

Many teenagers can subtract accurately using a number line. Some teenagers will remain dependent upon either lines or number squares for remembering where they are counting from or to. Practising counting forwards and backwards with number steps, number lines, and apparatus of various types will help teenagers understand the difference between numbers. All activities that help to develop fluency and confidence with the number system, backwards and forwards, with numerals, grids and by rote will make completing ‘taking away’ or subtraction sums easier to achieve. Knowing number facts will help teenagers to add and subtract more easily, and be able to check their answers.

For using subtraction in problems, teenagers need to learn that the order in which the numbers are written is very important, and that the biggest number comes first in a written sum.

Multiplication

Teenagers with Down syndrome can learn the same strategies as other teenagers for understanding and using multiplication. For example, strategies that can be taught and used for 3 x 2 include:

• Repeated addition: 2+2+2
• Counting in 2’s: 2, 4, 6
• Rules: 2 x 3 = 3 x 2
• Derived facts: 2 x 2 = 4, 3 x 2 = 4 + 2
• Fact retrieval: 3 x 2 = 6

Children can learn about multiplication as repeated addition from written sums, e.g. 4 x 2, 7 x 1, by laying out appropriate apparatus. For this method to be successful with Numicon shapes or Cuisenaire, teenagers must have completed the structured stages using activities supplied with the apparatus, so that they can recognise the images and their number meanings automatically.

Many teenagers find it helpful to learn multiplication facts, so they can be retrieved easily. Difficulties are most likely to arise when teenagers need to understand and organise a problem before using the known facts to solve the problem.

Translating practical problems to ‘equations’

Problems in words can be transferred on paper as a mixture of words and pictures or symbols to help the teenager understand the task. Teenagers can translate the written problem into a picture supported task, with help as necessary, and then associate the picture supported task with the numeral supported task (see [Figure 27]). Similarly, they could lay a problem out on a table using characters drawn on paper, or words, and use Numicon shapes to place with each character, picture or word. The shapes can then be collected together and used to work out the answer.

! A problem translated into pictures, words and numbers

Figure 27. A problem translated into pictures, words and numbers

When teenagers know the steps involved in multiplication they can learn to apply their skills to functional activities, such as understanding and adding coin values ( [Figure 28]).

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Figure 28. Demonstrating a multiplication strategy for life skills

Multiplication using a calculator

When multiplication is understood, so that the teenager knows the procedure and the sign, a calculator can be used to find the answer.

Teenagers will benefit from clear steps and repetition at each step. Teachers can create their own resources to provide the necessary practice at the right level for the individual.

To use a calculator may require a lot of practice. Some of the stages involved are described below.

Learning to use a calculator

For example:

1. How to turn on and off.
2. Number recognition of numbers on key pads.
3. Number recognition of numbers on digital display.
4. How to press a number only once (fairly quickly and lightly).
5. Understand the phrase ‘to enter a number’ or ‘have you entered..?’
6. Recognise and name the key pad signs for +, -, x, ÷ and =.
7. Know that the decimal point key is very important. It can change the number totally so should not be used accidentally.
8. Understand that the calculator ‘keeps’ or ‘remembers’ all the numbers that have been entered (since the calculator was last cleared), but only shows the number that was last entered.
9. Know that the Clear button (C) and the ‘off’ button clear the calculator.
10. Know to clear the calculator before using it for a new problem.
11. Know how to enter numerals in the same order as they are written, from left to right.
12. Check that understanding of place value with the calculator display is the same as with that for numbers written on cards or paper, by reading, writing, entering and naming written numbers that correspond with those on the display.

When basic calculator skills are mastered, teenagers can practice entering, reading the answer and writing it down for:

1. Addition by entering numbers to 9
2. Addition by entering numbers to 99
3. Subtraction by entering numbers to 9
4. Subtraction by entering numbers to 99
5. Addition by entering numbers to 999
6. Multiplication by entering number to 9
7. Multiplication by entering numbers to 99
8. Division by entering number to 9 and then 99

Teenagers need awareness of the ‘decimal point’ (what it looks like, what it is called) so that they are not misled when they practice division using a calculator and obtain an answer with a decimal, e.g. 7.5.

1. Awareness of ‘decimal point’.

As teenagers’ number and calculator skills improve they will learn more about the decimal point and its significance for money. Using a calculator is a good way to learn about decimal points. Learning how to add and subtract, read and name money values on a calculator is a particularly useful skill for young people with Down syndrome to master.

1. Can enter money values - either by omitting the point and working in pennies, or by placing the point correctly (till receipts can be used to practice this skill).
2. Can read and write numbers displayed on the calculator, and place point correctly (if used).
3. Can interpret the written number as a money value.
4. Can translate a money value into a number, ready to enter.
5. Can add values together and obtain an answer, e.g. from a shopping list, independently.

Multiplication using a framework

Teenagers can practise creating their own sums, using a framework with blank spaces (see [Figure 29]). A framework can be laminated and used over and over again.

Numbers and word cards can be held in place using ‘Velcro’.

Word cards can be varied, for example, fruit, sweets, money, cars, balls, etc. and ready prepared on word cards, in the same way as the numerals on numeral cards.

! Example of a multiplication framework

Figure 29. Example of a multiplication framework

Learning tables

Knowing multiplication tables is essential for working out answers quickly. The 2, 5 and 10 times tables are useful for many daily activities. Other tables are less important for daily use, but teenagers will benefit from learning them as automated facts.

Division

Sharing

Teenagers will benefit from sharing out items equally, or breaking up ‘whole’ numbers into ‘equal’ parts. Fractions can be introduced by sharing cakes, oranges, etc. and with worksheets (see Figure 42). ‘Sharing’ is the practical application of division, and can be reinforced in the classroom, kitchen, meal table and many other social situations. The vocabulary for division should be emphasised and repeated in discussion with teenagers with Down syndrome, for example, ‘whole’, ‘parts’, ‘group’, ‘share’, ‘split’, ‘separate’, ‘divide’ and ‘division’.

The formal strategies for division are similar to those used in multiplication.

For the problem 15 ÷ 5, teenagers may use:

• Knowledge of multiplication: 3 x 5 = 15
• Knowledge of addition: 5 + 5 + 5 = 15
• Derived facts: 2 x 5 = 10, + 5 = 15
• Fact retrieval: 15 ÷ 5 = 3

! When the language and symbols for division have been learned, a framework can be used to practice division, similar to the example shown for multiplication. Cuisenaire and other apparatus can be used to show division visually, to illustrate problems, e.g. 10 ÷ 2, 20 ÷ 4.

Division has extra difficulties with language compared to multiplication: the answer for multiplication is the same whichever way the numbers are placed, but this is not so for division. Language such as ‘divide X by Y’, ‘divide X into Y’, ‘X is divided by Y’ is hard for people with language and short-term verbal memory difficulties to follow.

Odd and even numbers

Teenagers will understand the significance of odd and even numbers when they learn to share and divide. Odd and even numbers to 10 or 20 can be learned by memory through games with numeral cards, with words on the reverse reading ‘odd’ or ‘even’. This knowledge will help them to understand ‘odd’ and ‘even’ through other games and activities. Earlier work with Numicon shapes will have helped to develop an understanding of ‘odd’ and ‘even’ numbers, as this feature of number can be seen, felt and experienced through addition and subtraction, multiplication and division games.

! Problem solving with help from drawings

Figure 30. Problem solving with help from drawings

Problem solving

Teenagers can be helped to understand mathematics problems presented in words by drawing a representation of the problem with them (see [Figure 30]). They can make lists of the important points and relate them to the picture. Just as flowcharts, lists, word webs, spider diagrams, pictures and concept maps can help teenagers with Down syndrome to understand text or remember complex information, so can these methods be used to help them understand number problems, so that they can more easily decide how to solve them.

Most teenagers with Down syndrome in the 11 - 16 age range will need help and support to understand problems and rearrange them so that they can more easily understand and solve them. However, the teenager’s own ideas should be listened to carefully and patiently, and encouraged as part of a ‘work routine’, to help develop independent thinking and problem solving skills.

Measurement information

The decimal system

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Measurements for length, weight and volume use the decimal system. Teenagers will learn the terms and abbreviations for each and the importance of their names, for example, centimetres and kilometres, grams and kilograms and millilitres and litres. This will be easier for teenagers who understand the decimal number system, as they can learn the word for each order of 10, (e.g. 10, 100, 1000; g, kg; cm, m, km; ml, l) through visual games and flash cards. Teenagers should learn that the words that follow a number really matter and that they must be found on the measure or scale they are using if they are to follow instructions for measurement correctly, using scales, rulers, jugs and cylinders.

Time

Time cannot be ‘seen’ to be measured in the same way as length and volume, or felt like weight. The measurement uses base 60 for changing between seconds and minutes and minutes and hours, and then 24 for hours in a day, 7 for days in a week and so on. This different system of measurement will have to be learned, by rote and by experience. Being able to read time and communicate about it is important in every day life and independent self-management in later life will require this skill.

Time is divided below into understanding the passage of time and planning ahead over days and weeks, learning to tell the time, and understanding short periods of time.

Understanding the passage of time over days, weeks and months

Teenagers are learning about time as we talk about everyday activities, use tenses in our language, and words that mean before, after, morning, afternoon, evening, today, tomorrow, yesterday, last week, next week, days of the week, seasons and months of the year.

Using a calendar, timetable or ‘time line’ to learn about time

Calendars and timetables, that include written words and pictures for regularly occurring events, help teenagers with Down syndrome to link ideas about time to real, meaningful things they can see and experience. Home-made calendars can include words for days of the week, ‘morning’, ‘afternoon’ and ‘night’ and clock faces showing the times and words for important parts of the day, including ‘bed-time’. Many teenagers will be able to use typical school timetables in secondary school, and some will gain from symbols and pictures added to their timetable.

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Figure 31. A flip chart for school days

The complexity of the calendar will vary for each teenager. A simple or first calendar can have squares for the days of the week, labelled with the written word and a photograph for each day to separate the weekend activities (or days at home) from school days. This may be interesting if made as a ‘lift the flap’ chart, with the days of the week written on the outer flap ( [Figure 31]). The calendar can be made more complex by having symbols or pictures and words for the separate activities of the days or evenings, when teenagers are familiar with how to use the calendar.

! A calendar for learning time words

Figure 32. A calendar for learning time words

Another type of calendar is illustrated in [Figure 32]. A pointer for ‘today’ can be moved along each day, and for older teenagers, ‘yesterday’ and ‘tomorrow’ pointers can be added. The authors suggest that a thick border is used between squares to show a ‘night time’ slot, labelled with words and a symbol or picture of a teenager sleeping, so night time is visual. Showing ‘night time’ as a slot becomes particularly helpful for understanding, counting and crossing off how many days, nights or ‘sleeps’ before a special event, like a birthday, holiday or other significant event.

Weekly and monthly calendars can be used to measure time for school projects that last a relatively long time, such as measuring plant growth, or waiting for frog spawn to change.

A simple 12 month wall calendar can be bought or made and used to mark annual events such as birthdays, family holidays and religious festivals such as Christmas. The use of a combination of day, week, month and annual calendars will give teenagers support for understanding varying lengths of time, linked to their own life experiences. These calendars can be used to talk about past as well as future events. Talking together about past events may be a valuable way to give a teenager a real sense of the length of 2 weeks or 2 months or 2 days, and practice at using the correct tenses. Calendars are useful should teenagers ask repeatedly when an event will be or if it will be that day: they can be referred to their calendar or timetable to check for themselves.

Personal event or ‘time diary’ with photographs and sentences

To help teenagers link themselves and their activities with time and time language, make books with the teenagers about their life events - a ‘time diary’. Parents may like to do this at home as well as staff at school. The task is made easier by using a Polaroid or digital camera to record real events that the teenager will remember and then to apply the language of time (the day, the time, how long the event lasted, when the teenager may do this event again) in sentences written to go with the photographs ( [Figure 33]).

! A time diary

Figure 33. A time diary

Teenagers can also practise telling, reading and writing the time using clock faces in the diary and their watches.

Using clocks to relate time to events during the day and to learn to tell the time

The time that teenagers have spent in school from 5 years of age offers many years for learning about time and associating times with the experiences of the day in a regular pattern. This will hopefully lead to teenagers being able to make judgements about the ‘feel’ or passage of time. For most teenagers, learning will continue into secondary school and for many years beyond this.

Within a whole-day time frame, teenagers will need to learn to tell the time, to know when events are going to happen, to organise themselves, to look forwards and know when things will begin or end. Being able to tell the time, or at least knowing the main times of the day when changes and breaks occur, helps to develop independence and a sense of security. Watching and experiencing the passing of time with an analogue clock provides a visual way of gauging time, to support the sense and feel of the passage of time. Teenagers can ‘see’ how much time must pass before a certain event. A very large clock is helpful for this, as the movements of the second and minute hands can be observed.

Teenagers will be introduced to learning to tell the time from a clock face. They may be helped to learn by having their own large clock with moving hands. The vocabulary for ‘o’clock’, ‘half past’, ‘quarter to’ and ‘quarter past’ can be written onto the clock face and word cards so that they can practice reading and saying the ‘time’ paired with reading the numbers and hands on the clock face.

For learning about the times for events of the day, teenagers can have their own visual timetable. It can show a series of clock faces with the times marked on them and a written or pictorial description of the event next to each clock face. At points during the day they can be asked to check their clock face for the next event to the clock on the wall in the classroom, to see if it is time for the event. If they cannot read the clock on the wall they should have a clock that they can read to match to their own clock face(s).

Teenagers can learn the times of the days when things happen to them - the time they have breakfast, go to school, have lunch, go home, have tea, go to bed, etc. with increasing complexity as they get older and as their skills become more advanced. Similarly, at home teenagers can learn the times in words and to recognise times on clock faces for their favourite television programme or other activities.

Teenagers will also need to read digital clocks, and while it is easier to read the number from a digital display it is much more difficult to learn about the measurement of time from a digital clock than from an analogue clock. With digital displays, as with analogue clock faces, two displays can be used, one showing the time of particular interest when an event will happen, and the other the actual time, so that teenagers can make comparisons between the two.

Teenagers need to learn the shorthand way of writing times, and when their number skills are sufficiently advanced, the 24 hour clock.

Wearing a watch helps teenagers to be more aware of the time, check times for events and practise telling the time. Teenagers who can count in 5’s and 10’s will be able to ‘tell the time’ approximately, and can work towards telling the time more accurately as their skills advance. Large watch faces, faces that have minutes marked and 10’s or 5’s marked will make this easier. Watches with an outer, moveable bezel show 10 and often 5 divisions, so teenagers wearing these do not need to be able to count in 5’s and 10’s independently.

Reading off the total number of minutes on an analogue watch or clock will help with time planning as times are usually written in this way. Unfortunately times are not spoken in this way: beyond 30 minutes times become something to the hour and the counting is different. This is a difficult idea, as is judging where the minute hand is for the closest 5-minute block, or describing times as ‘nearly’ or ‘just gone’ something.

Understanding short amounts of time

Measuring time, by timing an event, is a skill that has many practical uses, from understanding when there are ‘’5 minutes more to go’’ or ‘’how much you can do in five minutes’’ to measuring time when cooking. Teenagers using a clock to time can place markers on the clock face, so they can see where they are starting from and when the timed period will be complete, for example, 20 minutes for cooking cakes.

Timers can be easier to use than clocks for timing short periods of time. Timers are available for a variety of times and are helpful for showing time for completion of an activity.

Teenagers can also use analogue display timers, such as cooker timers, where time is counted down (so that the timer displays the amount of time still to go) and an alarm sounds when the time is up. Similarly, digital timers, usually on microwave ovens, go backwards (they provide meaningful practice for counting backwards) and make a sound when the timed period is complete.

Teenagers should use the timers and clocks in real situations with emphasis on life skills, especially meal preparation and cookery, at home and in lessons at school.

Money

Beginning to understand money

Pretend shopping games are helpful for teaching teenagers about the exchange of items and the purpose of money. They also help teenagers to understand the decimal number system and shopping games should be used in the school curriculum from 5 years of age.

! Item and cost cards

Figure 34. Item and cost cards

Teenagers in the 11-16 age range should also handle real money, help with paying for goods and have their own money to take out with them. Teenagers will learn to be responsible about money in school and outside of school by handling it. Many teenagers, as they become older, regard having money as a valued responsibility, like having a locker or door key when aged over 11. They will find it very hard to learn about money from the classroom experience alone, although learning number skills at school is an essential part of understanding how to understand money sufficiently to shop independently.

Teenagers first learn to recognise coins and notes by name. The numbers on coins are small so teenagers may be helped by having numbers stuck on coins, or cards that go with the coins that show the number in digits and the name of the coin. Teenagers can play matching games, for example, matching coins to the same coins, as they hear the name spoken, and matching coins to a coin card where the number and written name are also shown.

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Figure 35. Age-appropriate items

Teenagers can begin to find amounts of coins to add to make a value or cost for an item, although until they have sufficient number skills this is likely to be in 1p amounts.

This will not prevent teenagers from finding whole coins to match to costs though, such as 50p or one pound. Worksheet examples or objects for teaching money at school should use real amounts and age appropriate objects, rounded up or down to a suitable whole coin or note value ( [Figure 35]). Teenagers remember what they learn at school and to practice buying items for amounts that are not realistic is not helpful. Immersing teenagers in the concept by creating real, or if not real then meaningful, activities which enable them to use money at school as well as at home and in the community, will be more helpful for teaching money skills.

Materials for games to teach realistic values and coinage might include:

• Blank cards with logos, labels, pictures and drawings of items with their cost on one side, and the coinage required on the reverse. Teenagers can practise finding the correct money for the item. The game can be reversed so that, after using the cards, they guess the item from the coins and money shown on one side of the card, then turn the card over to check ( [Figure 34]).
• Pictures of items, with written word cards and money values on separate cards, to be read and paired or matched to the item. After practice, just the pictures of items can be shown, to see if the teenager has learned the values or can make approximate estimates ( [Figure 35]).

Expensive items (for example, mobile phones, phone cards, club football shirts, jeans, trainers, CDs, videos, and computer games) can be used with medium priced items (deodorant, shampoo, sanitary towels, make up, washing powder, meat, cheese, bus fare, cinema ticket, taxi fare, sweets) and items costing less than one pound (beans, newspaper, crisps, fruit, bread, canned drinks), so that a range of costs are represented. These can be paired with vocabulary and prices (see vocabulary lists) for ‘cheap’, ‘expensive’, ‘dear’ and ‘too much’. A pocket money game, with pocket money of for example £5.00, can be played with the cards.

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Figure 36. Teaching the value of 2p using coin cards

Learning how to add money, even to add 2p coins, will depend upon the teenagers’ progress in learning about the number system and addition. If teenagers do not understand cardinality and place value, trying to add coin values together will be very difficult. Even with these number skills teenagers are likely to need some extra cues to remind them, for example, that a 2p coin means 2 x 1p. They may be helped by putting larger numerals on the coins, using coin cards with the coins on and values drawn larger, or having dots added to remind them of the amount the coin represents ( [Figure 36]). If Numicon has been used, pictures of the coloured shapes drawn small can be used with the coins, or shapes can be stuck onto coin cards. Coins can also be placed onto a number line, and matched to numerals and Numicon shapes to help teenagers realise that the number on the coin tells us their value or worth, not their size, colour, shape or other features. Games of exchange will be an important aid to understanding the relative value of coins.

A wall chart (see [Figure 37]) can help teenagers to remember coins and their relative values. When teenagers have understood one coin value that is more than 1p they will be likely to understand other coin values more easily, and can practice making equivalent amounts using different coins. Teenagers can also use a ‘feely bag’ or a coin recognition game, beginning with two coins and then adding more. When coins have been guessed correctly, they can be placed on a board: the winner fills the board first.

! Wall chart showing coin values

Figure 37. Wall chart showing coin values

Teenagers’ knowledge of counting in 2’s, 5’s and 10’s can be reinforced and applied to counting out these coins of one type, as well as the more difficult task of adding amounts with different types of coins. Teenagers who understand place value with numbers may still be helped by revision of place value using £ and p symbols (i.e. £111 is £100 + £10 + £1). Pupils need to understand the significance of the decimal point, and a game with money cards showing decimal points will help them ( [Figure 39]). Games with a calculator are also helpful, as described earlier in the module.

Counting money and ‘changing’ it at a ‘bank’

Teenagers who know that one pound is 100p (or 100 pennies) can count money into ‘pounds’, making groups of 10p’s, 20p’s or 50p’s if these values are understood, as they count. The group of coins can then be exchanged for one pound coins. Working with money, according to each teenager’s level of skill, is a suitable homework or weekend activity. Changing money at a Post Office or bank, or in a slot machine (e.g. at an amusement arcade or leisure centre) could also be discussed and practised.

Sharing money - playing ‘banker’, ‘shopper’ or ‘shopkeeper’

A game where money is shared between people is an enjoyable way of practising money skills. Teenagers can combine this with role play, by stating how much they need for an item and writing it on an envelope. The ‘banker’ can then find the correct money for each person and place it in an envelope for them. Each person can then pretend to buy what they want at a shop. A ‘shopkeeper’ can check there is sufficient money for the items in the envelope. Each person can take a turn at being a banker, shopper or shopkeeper.

Real opportunities for spending and handling money, at school and in the community, should be used as often as possible. This should be planned into the teenager’s education plan, in consultation with parents.

Pocket money, saving and spending

Teenagers will be helped by receiving regular pocket money. If they receive money in units of one pound coins, they can save these and exchange them for £5 or £10 notes, before spending it. Discussion at home can also help teenagers learn the vocabulary and expressions associated with saving, shopping and spending behaviours (see Vocabulary list 2).

At school, teenagers can learn about money gradually, through a structured course of lessons. Ideas for a series of lessons on money, where it comes from, where it goes and how to look after it, are illustrated in [Figure 38].

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Figure 38. Money: Where does money come from? Where does it go? How to look after our money. Each part of the diagram can be learned about in separate lessons, with activities to help understanding, and the whole picture can be built up gradually. Activities might be about the bank, savings accounts, the bank statement, wages, benefits, cheque cards and cash cards, bills and spending money. Explanations should be in clear language, with a printed record for the teenager’s work file, as shown at bottom of diagram

Giving the closest coin or note value in payment

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Figure 39. Money cards for matching and sorting games to teach the meaning of the ‘.’ and ‘p’ symbols

When teenagers can add various coin values together to make amounts, they are moving towards buying items in shops independently. They may use a strategy where they find the exact amount, which is possible with many combinations of coins or notes within the price range. They may use the strategy of rounding up to the nearest 100p or pound, and offer this amount. If they have done this correctly they need not be too worried about checking their change, as long as they receive some change.

The experience of ‘not having enough’ money for something that is wanted will help develop the idea of saving money. Similarly, counting money for an item in a shop will be encouraged if teenagers have experienced ‘not having enough’ as well as ‘having enough’. (Providing the extra money out of kindness or not being expected to pay the full amount will not develop independent money skills.)

For accurately checking change, teenagers need to be able to count-on from the cost of the item, the lower number, to the amount they gave, the higher number, and to remember the difference and amount. This is difficult to do mentally in a real situation, but steps can be made towards this through learning experiences, number skill development and frequent practice in the 11-16 age range.

Money checklist

• Coin recognition by name
• Say names
• Games with pennies for small amounts to 5, then 10
• Match amounts with 1p coins to a number line
• Know how to add small numbers and count-on
• Know 2p = 2 x 1p
• Know one pound coin = 100 pennies or 100p
• Count in 2 p’s (2 x table, counting in two’s)
• Understand 10p - count in 10’s, 10 x table

! ‘Not enough money’, ‘what do you need?’

###### Figure 40. ‘Not enough money’, ‘what do you need?’

• Understanding place value to 100
• Understand 5p - counting 5’s, 5 x table
• Know that 10 x 10p’s =one pound, =100p
• Know that 2 pound coins = 2 x £1.00
• Understand written forms and decimal notation
• Know how to enter money values on a calculator and how to add and subtract them
• Know that notes are worth more than coins (as single items)
• Know that £5 note is less than £10 note, £10 note is less than £20
• Select coins and notes as needed
• Know about the existence of £50 note

Social learning

In the classroom

• Target language teaching for mathematical vocabulary

• Use supports and props to aid or reduce requirements on working memory

• Consider grammatical complexity of problems, instructions and new information - re-interpret, rewrite and explain to the pupil

• Help the pupil to avoid failure and develop success - be aware of pupils’ sensitivity to failure

• Apply understanding to age appropriate interests and for life skills

• Develop time and money skills throughout education

• Value familiarity with the numbers of daily life

• Have support available for the individual’s learning needs

• Know the price of some everyday items that interest the individual

• Use vending machines

• Know about pocket money and when it is received

• Know the vocabulary associated with saving, shopping and spending

• Have a bank account - know how much is in it

• Know about cheque books and payment cards

• Understand about shops - how they work, how to behave in them

• Understand the ‘buying’ behaviour for shops- queuing, paying, change, leaving

• Understand wages and earning money

• Understand saving

• Managing own money

Learning in the classroom and at home

Individual targets and focused teaching for number, money and time

Teenagers in secondary schools will be following programmes for developing number skills and learning about money and time, with activities and teaching materials designed to support each of their targets. They may learn as part of a small group with group teaching, or they may be following an individual programme in a classroom with peers of mixed abilities. Teenagers working at different levels from their peers can be included in maths lessons with their peers, with parts of the lesson differentiated, and part of the lesson used to practise their individual programmes for number, time and money, and other types of improvement to language.

New vocabulary and wider maths curriculum

Provided the priorities for number, time and money are practised regularly, sharing activities with peers from a wider maths curriculum is often enjoyable and valuable, as well as more socially inclusive. Sometimes the individual programme and class activity can be integrated together, but this is likely to vary from lesson to lesson. Steady progress should also be expected in learning new mathematical vocabulary, understanding diagrams, geometry, measurements and other areas of work that the teenagers are included in, for example, fractions and algebra ( [Figure 41] and [Figure 42]).

Cross curricular targets

Opportunities for learning maths basic skills will occur in lessons across the curriculum. Clear targets, described in the teenager’s Individual Education Plan, can be meaningfully worked towards in many lessons - for example, technology classes (food, graphics, design, information technology), art, languages (e.g. times of the day, days of the week, vocabulary - can be practised in English as well as French or German), geography (many forms of measurement, co-ordinates), history, English, drama, dance, physical education, as well as in maths lessons and through daily life skills such as buying lunch.

Shared targets, support and practice at home

Targets and their integrated use at school should be shared with teenagers’ parents so they can be integrated into daily life at home too, for example:-when watching television (competitive games and TV shows that use numbers - there are many of these), shopping, cooking, learning about buses and other forms of transport, using vending machines, reading petrol pump displays, collecting pocket money, saving and spending.

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Figure 41. Work of a 13-year old Italian pupil. [12 (Reproduced with permission)

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Figure 42. Participating in class work on fractions

Teaching new concepts through matching, selecting and naming

1. Start with matching
   The teenager is asked to match by putting the object, picture or card with the one that is the same. This is the step in which you are teaching the new concept so it is important to use the appropriate language e.g. ‘’This is a 50p coin, can you put it with the other 50p coin’’. Once the teenager can match items correctly, move on to selecting.
2. From matching to selecting
   The teenager is now asked to select each of the items by name e.g. ‘’Can you give me (or show me) the 50p coin?’’ Once the teenager can demonstrate correct comprehension of the words by selecting the items correctly, move to naming.
3. From selecting to naming
   The items are named accurately on request. Extra practice at naming functional items may be necessary to improve speech clarity, e.g. ‘’one pound coin’’.

Teaching methods and supports for language and memory

Structured teaching methods

Structured activities that include errorless learning, such as matching and selecting games (see box) can help many teenagers with Down syndrome to learn new ideas, for example, coin recognition, coin values, relative value, times on clock faces, symbols and their names and meanings. Any materials that teenagers enjoy can be used as part of a teaching and learning game. Structured games can teach one particular part of a skill that can then be built on by learning the next part in later games. In this way, teenagers can progress in small steps until they learn the whole skill successfully, without being overwhelmed by too much new information or too many differing task requirements at the same time.

Breaking the task into small structured steps usually helps young people to do things for themselves; they need less explained to them and can focus on completing the task, without having to process spoken information simultaneously. It is important to model activities and to use clear, uncluttered and attractive resources. The easier it is for teenagers to ‘see’ how to succeed, the more likely they will copy and engage in the tasks with enjoyment. Many new skills can be learned through modelling, ‘turn taking’ and group activities.

Using computers

The computer is a valuable aid to learning for teenagers with Down syndrome. The computer enables them to use their strengths in visual learning, as information is always visual on the screen, and their strengths in being able to choose the right answer by pointing (using the mouse). Computer activities are usually highly motivating and can bring a fresh approach to number learning. Programs for basic number, time and money skills are available, such as Number Shark, Mathsbook, Talking clocks, Money Talks. TODO: references 15

Supporting language learning and understanding

• Write vocabulary on blank card
• Illustrate meaning on the back of the card
• Re-write text book sentences and break them down
• Write instructions in smaller amounts, in a list

Written supports for teaching vocabulary, remembering sentences and following instructions

Reading words at the same stage that teenagers are developing their vocabulary understanding will help them to remember new vocabulary and meanings. Single words can be written onto flash or word cards and symbols or pictures can be added to help illustrate the concept. For teaching comparatives, cards for ‘bigger than’ and ‘smaller than’ can be arranged with objects at either end. Reading number words will help the teenager to distinguish between words that sound similar when they hear them and that they may not be discriminating between, for example ‘teens’ and ‘ty’ words, such as ‘fifteen’ and ‘fifty’. These words can be read and spoken aloud with different points of stress on the final syllables.

Sentences can be written onto strips of paper or card to help teenagers remember what they are learning. Complex sentences can be broken into simpler sentences to help teenagers follow the steps required more easily.

See also:

• Memory development for individuals with Down syndrome

Teaching materials and books, even for teenagers who are delayed in learning maths and numeracy, do not necessarily allow for language delay so the language used may prevent the teenager from demonstrating their understanding and skills. Card and paper should be ready for support staff to write extra prompts or information that allow the teenager to work more independently, rather than constantly needing verbal translation and reminding by an adult or other person. This type of translation may also be needed to support learning and activities on the computer as well as work with objects, work sheets or text books. Instructions can be written in a list, to support each step in the task, and teenagers can tick or cross the steps off as they are completed.

Memory training: Rehearsal and repetition

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Figure 43. Lift-the-flap charts to learn the order of numbers 1-5 and to count in 5’s

The rehearsal technique which is used in memory training activities can be used to teach teenagers number facts ( [Figure 43]). The rehearsal technique is explained in full in the memory module in this series. The teenager learns the items in order by closing all the flaps then lifting the first one, naming the item, closing the flap and asking the teenager to name the item (now hidden). Next, the first and second items are uncovered, named, covered and the teenager is asked to recall the 2 items. The learning continues in this way until the teenager knows all 5 items. Consider applying this technique to learning numbers in order, counting in two’s, five’s and ten’s, days of the week, months of the year, telephone numbers and aspects of time, linked with photographs.

Maths practice

Teenagers will gain from daily practice to learn new skills and to revise previously learned information and skills.

The following information comes from Sheila Hutchins, TODO: references 13 who describes her work supporting a teenager in maths lessons (see [Figure 44]).

This work is broken down into mixed maths. This gives Joni reinforcement everyday, for the maths she has learned. By breaking it down into small parts it keeps her interested.

Monitoring progress is very important and by giving her maths in this way monitoring can be carried out daily. Most days Joni can do most of her maths independently with the exception of long multiplication, which has only recently been introduced. Although Joni can sometimes work independently her mind sometimes wanders off onto other things and this is when she needs words of praise and encouragement to bring her back to the task.

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Figure 44. ‘Mixed maths’ for daily practice

Teenagers will need their maths resources available to them for daily use, for example a ruler, number cards, apparatus e.g. Numicon shapes, calculator, number lines, blank card (for new vocabulary), word box for word cards, time diary and a memory training game.

Unless skills are practised often they may not be maintained or developed further.

Linking classroom work to the community

Local daily newspapers show advertisements for items for sale, local entertainment and events. These can be found and discussed, with costs and fees.

Leaflets show forthcoming activities and costs, and can be collected from cinemas, sports and leisure centres, the civic offices and local colleges. These will lead to discussion of present and future leisure skills and expectations for adult life, as well as helping to develop an understanding of the necessity of money for accessing these. Many towns, cities and regions have ‘leisure cards’ that reduce the cost of activities to people on a low income, and these can be explained and discussed.

Summary

Number skills develop through the mastery of significant stages that are the same for children with and without language and cognitive differences. Teenagers with Down syndrome can continue to develop new skills and master the significant stages that are learned by pupils who do not have learning disabilities in their primary school. It is essential for teenagers and young people to continue in their development of basic number skills so that they can understand and use money, time and other types of measurement in real situations, as a life skill.

Learning number skills requires a great deal of practice, even for the most able young people with Down syndrome, and many teenagers are willing to work hard to master new skills. Some teenagers enjoy maths, are self motivated and achieve higher skills. Most teenagers require graded programmes of work and need to practice number skills daily, with encouragement and reward, to further their attainments during their secondary years.

The more teenagers can see the relevance of their activities and learning to everyday life and independence, the more motivated they are likely to be to practise and learn. The more they are supported at home and across the curriculum at school to practise their skills, the faster they will progress.

There are no short cuts to mastering the stages needed to understand and use numbers, but there are teaching methods, materials and activities that can help teenagers with Down syndrome to learn, particularly those that use visual images to illuminate concepts, support memory and aid practise.

There are strategies and aids that can help pupils to manage practical number skills without necessarily understanding the numbers involved. The information, context and purposes of many of the numbers we encounter and use daily can be memorised, written down and learned through experience. The relative values of money and the coins needed for the purchase of particular items can be memorised by teenagers who cannot add or subtract. However, this approach is limited. For example, understanding place value for hundreds, tens and units helps with reading bus numbers, cooking and weighing. To operate a calculator requires place value skills and, in order to use a calculator for money, an understanding of the decimal point as it relates to money values is needed. Knowledge is required for teenagers to decide when they do not know (and thus to identify the need for help) as much as for when they do know and can succeed with the skills required of them.

The authors recommend that both basic and practical approaches to teaching are maintained for teenagers aged 11-16, in maths lessons.

• Development of number skills (to at least 100), using a numeracy teaching programme, (these can be transferred to practical skills fairly easily when mastered), and
• Learning basic practical skills for independence, including money, time and number recognition for weighing, measuring and other life skills. This requires that teenagers are given opportunities to practise and become competent and confident, with or without understanding of the numbers involved, such as buying their own lunch, receiving and spending their own money and cooking and preparing food.

We recommend that teachers and parents do not stop the first approach to focus exclusively on the latter in secondary school, as many teenagers have more mature work habits than primary aged children and are more likely to attend to teaching instruction and undertake the amount of practice needed to master number skills. Teenagers have 5 years between the ages of 11 and 16 when they may be at their most ready to learn - as much as possible should be made of this.

Summary: supporting learning

• Vocabulary - teach with word, picture, sign, symbol - use teaching methods that focus on vocabulary learning, and vocabulary checklists
• Grammar - use reading, check comprehension, rewrite into separate sentences
• Rehearsal - memory training techniques, especially for number and time sequential information, for example, 11 - 20, counting in ten’s, five’s and two’s, days of the week, months of the year
• Help with recording
• Help the pupil to ‘see’ quantities associated with numbers to understand them, and to see and use the patterns in the number system
• Extra materials available for pupils - box, digits, lines, 100 table, numerals, ‘Numicon’ shapes and or other apparatus being used by pupils to support their learning, extra blank card and paper for creating new support materials, written and picture prompts
• Applications to everyday life
• Pupils will continue to learn more with increasing age, appropriate teaching and opportunities for practice
• Collaboration with parents so that daily experiences are used to maximise learning

Vocabulary lists for numeracy

See also:

• Vocabulary checklists and record sheets: Checklist 1 - First 120 words
• Vocabulary checklists and record sheets: Checklist 2 - Second 340 words
• Vocabulary checklists and record sheets: Checklist 3 - Third 350 words

[these links will take you to the relevant product page at the DSE International Online Shop]

A structured approach to teaching number skills is helped by knowing the vocabulary for maths that typically developing children learn as they progress through school. Early number, time, place and attribute words extracted from Down Syndrome Education International’s vocabulary checklists are shown below. If your teenager has limited vocabulary, the checklists are useful tools for setting targets and monitoring progress in language learning.

• Checklist 1 - First 120 words

• Time: again,

• Quantity: all gone, more

• Place: down, in, off, on, there, up

• Attributes: big, little

• Checklist 2 - Second 340 words

• Number: One, two, three, four, five

• Time: day, later, morning, night, now, today, tonight

• Place: away, back, here, inside, out, under

• Attributes: shape, circle, square, triangle, size, small, blue, green, red, yellow, all, another, empty, more, none, some, time, again, same

• Checklist 3 - Third 350 words

• Number: Six, seven, eight, nine, ten, eleven, twelve, numbers

• Time: after, afternoon, before, minute, next, once, time, tomorrow, yesterday

• Place: about, above, around, at, behind, beside, by, first, from, in front, into, last, next to, on top of, over, through, to, with

• Attributes: fat, heavy, light, long, tall, thick, thin, tiny, short, black, brown, orange, pink, white, any, empty, full, half, many, much, a bit, a lot, different, each, every, lots, some

Many other words that will be used in school, including instructional words to help teenagers participate in activities, are presented in the vocabulary boxes.

Many more words are used in schools and a comprehensive vocabulary checklist for numeracy (designed originally for children in primary schools) is available at The National Strategies Site, a government website for teachers at http://nationalstrategies.standards.dcsf.gov.uk/node/84996

The authors have separated two sets of vocabulary ( [Set one] and [Set two]) to help teachers and parents to plan and assess language teaching in the classroom and at home. Set two includes and expands on the vocabulary items in set one.

Vocabulary list - Set one

same number/s
different number/s
number line
dice
dominoes
pegs, peg board
zero, one, two, three… to twenty and beyond
none
how many…?
count, count (up) to
more, less
how many times?
pattern,
pair
‘teens’ number
the same number as, as many as
altogether
one more, take (away),
leave
the same as
| | Measures | | size, big, small
enough, not enough
long, short, tall
wide, narrow
deep, shallow
thick, thin
ruler
weigh
heavy/light
full
empty
holds
container
| | Time words | | birthday, holiday
morning, afternoon
night
bedtime, dinnertime, playtime
today, yesterday, tomorrow
before, after
first, next, last
now, soon, early, late
quick
fast
slow
old
new
hour, o’clock,
clock, watch, hands
once, twice
| | Shape words | | shape, pattern
flat
straight
round
corner
circle
triangle
square
rectangle
star
hexagon
diamond
| | Position, direction and movement | | in, on
over, under
above, below
top, bottom, side
on, in
outside, inside
around
in front, behind
front, back
before, after
beside, next to
middle, edge
left, right
up, down
forwards, backwards
through
to, from, towards, away from
movement
slide
roll
turn
stretch, bend
number
| | Money words | | money
coin
penny
buy
spend
pay
change
names of coins (1p, 2p)
| | Instructional words | | match
listen
give me
join in
your turn, my turn,
say
remember
start
look a
point to, show me
put
find
choose
make, build
tell me
read
finish, end
count
answer |

Vocabulary list - Set two

think, imagine, remember
start with, start at, look at
point to
put, place
arrange
rearrange
change, change over
split
separate
carry on, continue
repeat
what comes next?
find
choose
collect
use
make
build
tell me
describe
pick out
talk about
explain
show me
read
write
record
trace
copy
complete
finish, end
fill in
shade
colour
tick, cross
draw
draw a line between
join (up)
ring
arrow
cost
count
work out
answer
check
| | Addition and subtraction | | add, more, plus
make, sum, total, altogether
score
double, near double
one more, two more… ten more
how many more to make…?
how many more is… than…?
how much more is…?
subtract, take (away), minus, leave,
how many are left/left over?
how many are gone?
one less, two less, ten less…
how many fewer is… than…?
how much less is…?
difference between
half, halve
is the same as, equals
sign
| | General | | same number/s
different number/s
missing number/s
number facts
number line, number track
number square
number cards
abacus
counters, cubes, blocks, rods
die, dice
dominoes
pegs, peg board
same way, different way
best way, another way
in order, in a different order
| | Measures, shape and space | | Measures (general)
measure
size
compare
guess, estimate
enough, not enough
too much, too little
too many, too few
nearly, roughly, close to, about the same as
just over, just under
| | Length | | length, width, height, depth
long, short, tall
high, low
wide, narrow
deep, shallow
thick, thin
longer, shorter, taller, higher… and so on
longest, shortest, tallest, highest… and so on
far, near, close
metre
ruler, metre stick
| | Mass | | weigh, weighs, balances
heavy/light, heavier/lighter, heaviest/lightest
weight, balance, scales
| | Capacity | | full
half full
empty
holds
container
| | Time | | time
days of the week: Monday, Tuesday…
seasons: spring, summer, autumn, winter
day, week, month, year,
weekend
birthday, holiday
morning, afternoon, evening
night, midnight
bedtime, dinnertime, playtime
today, yesterday, tomorrow
before, after
next, last
now, soon, early, late
quick, quicker, quickest, quickly
fast, faster, fastest
slow, slower, slowest, slowly
old, older, oldest
new, newer, newest
takes longer, takes less time
hour, o’clock, half past
clock, watch, hands
how long ago?
how long will it be to…?
how long will it take to…?
how often?
always, never, often, sometimes, usually
once, twice
| | Shape and space | | shape, pattern
flat
curved, straight
round
hollow, solid
corner
point, pointed
face, side, edge, end
sort
make, build, draw
| | 3D shapes | | cube
cuboid
pyramid
sphere
cone
cylinder
| | 2D shapes | | circle
triangle
square
rectangle
star
| | Patterns and symmetry | | size
bigger, larger, smaller
symmetrical
pattern
repeating pattern
match
| | Position, direction and movement | | position
over, under, underneath
above, below
top, bottom, side
on, in
outside, inside
around
in front, behind
front, back
before, after
beside, next to
opposite
apart
between
middle, edge
centre
corner
direction
journey
left, right
up, down
forwards, backwards, sideways
across
next to, close, far
along
through
to, from, towards, away from
movement
slide
roll
turn, whole turn, half turn
stretch, bend
| |
| | Numbers and the number system | | Counting, properties of numbers
and number sequences | | number
zero, one, two, three… to twenty and beyond
zero, ten, twenty… one hundred
none
how many…?
count, count (up) to
count on (from, to)
count back (from, to)
count in ones, twos… tens…
more, less
odd, even
every other
how many times?
pattern, pair
| | Place value and ordering | | units, ones
tens
exchange
digit
‘teens’ number
the same number as, as many as
equal to
| | Of two objects/amounts | | more, larger, bigger, greater
fewer, smaller, less
| | Of three or more objects/amounts | | most, biggest, largest, greatest
fewest, smallest, least
one more, ten more
one less, ten less
compare
order
size
first, second, third… tenth, eleventh… twentieth
last, last but one
before, after
next
between, half-way between
| | Estimating | | guess how many, estimate
nearly, roughly, close to
about the same as
just over, just under
too many, too few, enough, not enough
Organising and using data
count, sort, vote
list
group, set
table
| | Vocabulary for solving problems | | making decisions and reasoning
pattern
puzzle
answer
right, wrong
what could we try next?
how did you work it out?
count out, share out, left, left over
number sentence
sign, operation
| | Money words | | money
coin
penny, pence, pound
price
cost
buy
sell
spend, spent
pay
change
dear, costs more
cheap, costs less, cheaper
costs the same as
how much…? how many…? total
|

References

1. Dienes, Z.P. (1964). Mathematics in Primary School. Melbourne: Macmillan. Materials now available from educational suppliers as Multibase.
2. Cuisenaire, G. and Gattagno, C. (1957). Numbers in colour. (3rd Edition) London, UK: Heinemann.
3. Numicon materials published by Numicon Ltd., available from The Down Syndrome Educational Trust.
4. Buckley, S.J., Bird, G., Sacks, B., and Archer, T. (2002). A comparison of mainstream and special education for teenagers with Down syndrome: implications for parents and teachers. Down Syndrome News and Update, 2(2),46-54.
5. Buckley, S.J., Bird, G., Sacks, B., and Archer, T. (2006). A comparison of mainstream and special school education for teenagers with Down syndrome: effects on social and academic development. Down Syndrome Research and Practice, 9(3), 54-67.
6. Wing, T. (2000). Serendipity, and a special need. Mathematics Teaching. 174, 27-30.
7. Geary, D.C. (1994). Children’s mathematical development: research and practical applications. Washington, USA: American Psychological Association.
8. Gelman, R. and Gallistel, R. (1978). The child’s understanding of number. Cambridge, MA, USA: Harvard University Press.
9. Irwin, K.C. (1991). Teaching children with Down syndrome to add by counting-on. *Education and **Treatment of Children*, 14(2), 128-141.
10. Thompson, I. (1997). Numbers fall into place. Times Educational Supplement, 3 October, Mathematics Extra.
11. Count Us In! The Questions Publishing Company Ltd. and Staffordshire County Council (1999).
12. Monari Martinez, E. (1999). Learning mathematics at school and…later on. Paper presented at ‘Joy of Living’ International Conference. Israel DSA. Jerusalem.
13. Personal communication.
14. Ewan, C. and Mair, C. (2002). Wiltshire Pilot Project - Numicon. Wiltshire County Council. Down Syndrome News and Update, 2 (1). 12-14.
15. These software programs are available from The Down Syndrome Educational Trust.

Resources

Many of the resources referenced in this publication are available via Down Syndrome Education International’s specialist mail-order service. For further information see http://shop.downsed.com

A range of number teaching resources including plastic place value arrows, number squares and lines, timers, clocks and number games are available from ASCO Educational Supplies Ltd., 19 Lockwood Way, Parkside Lane, Leeds LS11 5TH. (Some materials in this catalogue are only suitable for younger children, but others are appropriate for teenagers.) Tel: 0113 270 7070. Fax: 0113 277 5585. Email: tom@binder.tele2.co.uk

Acknowledgements

The authors would like to thank the teenagers, parents, learning support assistants, teachers and schools they have worked with in schools across the UK, whose talents have contributed towards this module. They would also like to thank Vikki Horner, Joanna Nye and Ben Sacks for their suggestions for activities and their helpful comments on various drafts of this module. However, the responsibility for the final content, and any errors, is solely that of the authors.

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.

The term ‘teenagers’ has been used as the generic term for this age group, even though the youngest are 11 year olds, as ‘children’ is no longer appropriate.

In this module, we have adopted a straightforward and direct style in which the reader is addressed in the first person. They have done this because the module is mainly concerned with practical activities and instructions, and we have found that this direct, active style has been appreciated by readers in previous publications.