Supporting the development of number and mathematics skills in primary education

Introduction

The mathematics curriculum during the primary school years focuses on giving children a basic set of skills for use in their daily lives as well as a foundation for understanding more advanced mathematics. It includes 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 child 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.

This module provides examples and practical ideas to help parents and teachers to teach children with Down syndrome. There is very little research into their number development but at present, the evidence indicates that they often find understanding number more difficult than learning to read. However, as in all areas of their development, there is wide variation in individual progress, with some children showing an aptitude and interest in number and learning at a rate within the range seen for typical children of their age and other children only mastering a simple level of counting by the time they leave school as teenagers. The children 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.

Experience suggests that number work in the classroom may become too abstract too quickly, as children are expected to be able to do sums with numbers on paper. Children 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 children 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 children at home and in school. Ideally teachers and parents will be working in partnership and parents can help children practice and generalise their classroom learning in real situations at home such as those which require counting or weighing or the use of money, and by playing number games with them.

The activities and strategies for supporting children’s learning are based on what is known about the learning difficulties of children with Down syndrome and what is known about how typically developing children learn number. Children with Down syndrome will come into school with significantly delayed language and therefore they may not have the basic vocabulary for number, size, colour, shape and quantity that will be used in the classroom. For this reason, a list of number vocabulary is included in the module. The children will have been making slower developmental progress and this may have resulted in less opportunity to play games which teach number concepts to them in their preschool years. The children usually experience delay in developing fine motor skills and this may lead to fewer opportunities to manipulate, sort and count small objects.

In the classroom, therefore, children with Down syndrome will need help to learn 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 and using number cards and number lines for example. None of these difficulties are unique to children with Down syndrome and many teachers will be using the same strategies to support learning for other children with speech and language delay, motor difficulties or memory difficulties, in the class.

Some children with Down syndrome will have some knowledge of counting, colours, size and shape at 5 years of age but many will be just starting to learn these ideas, therefore the first part of this module provides activities to teach basic concepts and then moves on to understanding number, calculating, time and money. It includes examples of children’s work and individual rates of progress, and provides guidance to targets for children with Down syndrome.

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 typically developing children, and the rationale for the practical approaches recommended.

Development of number skills for children with Down syndrome: examples of achievements

What skills might children bring with them to school?

Up to the age of 5 years young children with Down syndrome have learned about number and mathematical words through play, song, nursery school activities, home teaching and other life experiences. At 5 years of age, many are saying and trying to use some of the numbers from 1 to 10 in counting tasks when they begin school. They may have heard and seen higher numbers through supported counting games, in their environment and through conversation. Early learning activities, as for all children, include learning about the sequence of numbers in our number system, counting, and understanding of quantities. Sign language (using fingers) as well as materials and cards showing patterns and numerals, can help to compensate for weaknesses in speech, so a child does not need to be able to speak to be involved in number activities.

Young children with Down syndrome should experience typical mathematical language, especially words for number, in the same way as other children. Most 3 to 5 year olds with Down syndrome are capable of learning the stable order of numbers to 10, can learn to recognise numerals, develop one-to-one correspondence from games and counting exercises and can begin to learn about number by seeing different quantities. They may be beginning to link quantities they can see with number words and numerals around the time they start school.

Supported early learning activities are likely to have taught them the meanings of same and different, how to match, compare and sort, and may have introduced them to the numerals and sequence of numbers to 10.

Many children will be noticing how quantities differ, without linking this to a system of number. Some children have age-appropriate number skills: such as counting to 10, saying numbers to 15 or 20 and linking small quantities they can see with number words and numerals.

Some children will have experienced only a little number teaching before starting school, but will progress quickly in school with appropriate teaching and practice. While most children with Down syndrome at the age of 5 understand same and different, a small number of children may not understand these words and will be learning how to match in school.

What will they learn in school and how may they progress?

During the infant years, age 4, 5, 6, and 7, children will be working on developing number skills up to 20, with some knowledge of numbers beyond 20. At age 7 years some children will be working on numbers up to 5, while some will be able to count by rote beyond 20, read numbers from a 100 square, be able to add and subtract to 10, order numbers to 20 and count-on (i.e. if asked to add 4 and 2, the child can count on from 4, i.e. ‘’4,5,6’’, and does not need to start counting from 1 in order to carry out the task). Many will be joining in with activities in school, counting to bigger numbers, or counting in twos, fives and tens, first as a rote memory game and later as a series of mental additions. All will be continuing to learn new vocabulary. Learning about money, time and measurement will be part of the curriculum for all children in this age range, and children with Down syndrome can be included in whole-class teaching, with activities simplified and differentiated as necessary.

In the junior years (ages 8 to 11 in the UK), many children will know about numbers to 100, counting in ‘tens’, ‘fives’ and ‘twos’, addition to 20, subtraction, early multiplication and division. They will also have increased their mathematical vocabulary knowledge. Some children can add larger numbers using learned procedures, with visual and mental strategies.

During their junior years, more children 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 children will be adding and subtracting confidently, others will be carrying out learned procedures but may be easily confused by changes in style of presentation, materials or language used. Some children who enjoy maths and have progressed well 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 children to understand problems, use procedures and number facts, and read tables, graphs and grids. Some children in junior schools may not have mastered number to 10, although their skills will be gradually improving. Many children with Down syndrome enjoy maths, whatever level they are working at, and a small number of children are good at maths (e.g. functioning within the range of other children of similar age).

Teaching children with Down syndrome

Children with Down syndrome may need more teaching and practice in order to learn about number and maths than other children. They will also benefit from special consideration of aspects of their language and cognitive profiles, and how their learning strengths and weaknesses can influence their progress (see [box below]).

Activities, supports and teaching targets that will influence progress are listed below and will be developed further in the following sections.

• Experience of numeracy - at home and in school through social interaction, saying or ‘chanting’ numbers with others, counting, dice games, board games, wall friezes and displays, hearing number used during activities with everyday items and toys and through teaching games
• Motor skills - handling objects, construction play, 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 and daily routines, using fingers and other visual cues
• Learning to count - 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, and numerals for visual support

! 100 square

A ‘100 square’

• Using practical materials which represent the number system visually to support their learning
• Learning to recognise patterns, matching patterns and arranging items into patterns
• Games, for learning to recognise numerals by matching individual numerals to each other and matching numerals to a number line
• Using and playing with money
• Using daily and weekly calendars to develop understanding of time

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. Games and puzzles, where the rules can be picked up quickly by watching a demonstration, will also help children with Down syndrome to learn and understand mathematics.

Language for number

Early teaching activities

Teaching methods

There are many everyday activities where number and other attributes (e.g. colour, size, shape) can be introduced, practised and learned through play in the home and school environment. Structured activities that include errorless learning, such as matching and selecting games (see [box below]) can help many children with Down syndrome to learn new ideas. Any materials that children enjoy can be used as part of a teaching and learning game. Structured games are games that have been designed to teach one particular part of a skill that can then be built on by learning the next part in later games. In this way, children 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. Games where too much information is presented at once can leave children feeling that they have failed, and they may then not want to play that game again.

Breaking the task into small structured steps usually helps children 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 children to ‘see’ how to succeed, the more likely they will copy and engage in the tasks with enjoyment. Many number and maths skills can be learned through play and teaching games.

Matching games

Use matching games for teaching size, colour and shape names. Many types of visual matching games are useful, where the child is helped to place the ‘’same’’ with the ‘’same’’ and hears what it is called, by listening to the spoken word and seeing the sign. (Children with Down syndrome are often taught signs to support their spoken language development.) The stages in matching games are: a matching stage, a selecting stage and a naming stage (see box right).

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Figure 1. Bean bags for colour matching

This way of teaching is very effective as it supports the child to learn in an errorless fashion, succeeding at each step, and it can be used to teach a whole range of new concepts throughout childhood. Remember to prompt the child as necessary at each step to ensure that they succeed while they are learning.

Category words

It helps children to learn the concept if you use the category word - for example ‘’these are colours’’ or ‘’red colour’’ as well as the colour word ‘’red’‘, and similarly use category words for shape and size, e.g. ’‘What shape is this? Is it a square or a circle?’’ ‘’What size is this? Is it big or small?’’.

Coloured bean bags and a large piece of paper with coloured rectangles, or circles of colour with discs to match to them, make easy matching games in the early stages of colour name learning, when identical items are needed ( [Figure 1] and [Figure 2]).

In our experience, colour learning is often helped by giving the colour name in print.

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Figure 2. Circles for colour matching

When you know that the child understands the colour name and can demonstrate their understanding through selecting games, begin to sort objects or items that share the colour feature but differ in other features.

Combining attributes and ordering items

Logic blocks or similar educational materials are available in different shapes, colours and sizes and can be used to teach these concepts. They are also useful for teaching combinations of attributes in a sentence, by asking the child to select on two or three attributes at once (for example, ‘’where’s the big, red circle?’’) ( [Figure 3]).

Children may have difficulty remembering a request with three criteria to process, so it will be important to write out or to repeat the sentence while they do the task.

Ordering items requires a series of ‘comparisons of two’ to be made and involves looking, remembering and comparing skills, for example, for ordering in size.

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Figure 3. Blocks for teaching attributes

The number of items to be ordered can be gradually increased from three upwards. Items can be ordered on many features as well as size and number, such as the loudness of ‘noise’ the item makes, practising the vocabulary ‘’loud’’ or ‘’quiet’‘, or weight,’‘heavy’’ and ‘’light’’.

The language and ideas for comparing, called comparatives, can be introduced in ordering games, for example ‘’taller than’‘,’‘smaller than’‘,’‘heavier than’’ etc. ( [Figure 4]).

Extending children’s understanding of words for qualities or attributes of items (or people, animals or activities) beyond colour names, shape names, size (‘’big’‘,’‘small’’) and number will help them to think about the concepts.

Children will hear this vocabulary when they reach school if not before, and the more experience children with Down syndrome have, the faster they will learn.

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Figure 4. Dolls for ordering games

These and other words and concepts will help to give children more elaborate ways of comparing and thinking. They will also help to improve the ways in which they can categorise information and improve their memory and language skills.

Using computers

The computer is a valuable aid to learning for children 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). There are many good software programmes available to support early learning of colour, shape, size and colour. There are also many programmes for early and later number skills. TODO: references 4

Understanding number and mathematics: a formal approach

Primary skills list

The lists below for Number; Money; Time; Other measurement; Shape, place and data, summarise the skills and concepts that children with Down syndrome are likely to learn about in primary 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 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
• 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
• PPlace value, units, tens, hundreds
• Count, read, (write) and order numbers to 100
• Know 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 and then larger amounts
• £ and p notations
• Use 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 - second, minute, hour, day, week, month, year
• Tell the time

Other Measurement

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

Shape, place and data

• Shapes, square, circle, triangle
• Position, direction and movement
• Understand simple graphs
• More 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 children 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 children at this stage, or as quickly as they need to in real situations. But children 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 children with Down syndrome achieve more than this, and with their peers will learn additional skills, described below as ‘advanced’ for children 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

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

Learning about number to 10

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Figure 5. Numicon shapes and pegs

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.

Children begin to learn to count from an early age, and they should be encouraged to do so. However, learning to count will not necessarily teach children to understand the nature of the number system and there are additional activities that can be used to help them achieve this.

In the English numeracy curriculum there is a heavy emphasis on counting in the early years and on mastering mental arithmetic. Teachers of mathematics have differing views about the best ways to teach children to count and to understand the number system. Some argue that counting activities alone will not lead children to understand the nature of the number system and advocate the use of materials which provide a visual spatial representation of the system (such as Numicon and Cuisenaire) to help children. TODO: references 5 Numicon (see [Figure 5]) 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 5

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.

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Figure 6. Picture cards for matching and ordering

Drill and practice tends to be unfashionable but there are good arguments for suggesting that children should practise the count word sequence until it is mastered to an automatic level, and similarly 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 6

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

The visual memory and visual learning strengths of children 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’’ ( [Figure 6]).

Children 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 5]).

! Record of pattern and counting work

Figure 7. Record of pattern and counting work

Numicon materials are available for a single child or whole class use. TODO: references 1 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. Pegs are included for counting and pattern activities, and each shape has holes that the pegs can fit into. Matching the shapes to each other, selecting and naming them, associating numbers of items (the Numicon pegs and other items) with each shape, ordering shapes, associating numerals and number words with them, and finding the number and shape position on a number line, are all activities that will help to develop children’s understanding of number.

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 8]). These principles can be learned through structured games, including games with whole numbers, recognising patterns and other types of visual imagery.

More practice and explicit teaching may be needed to help children with Down syndrome understand each of these principles. Many children with Down syndrome in the 5 year age range have some understanding about one-to-one correspondence and stable order, although being asked to use both at the same time may be difficult, as in a task to count objects. As each skill becomes better learned, then the two can be used simultaneously to ‘count’.

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 8. The “how to count” principles - and steps in understanding number, based on Gelman and Galistel. TODO: references 7

Pointing at pictures

Children can be helped in the early stages of counting by counting items in pictures and on pages in books. They are not then required to hold or move an object, only to point once and move on to the next one. This game can be modelled and is copied easily by most young children with Down syndrome.

Keeping track

Children can be helped to keep track of procedures when counting objects, by placing items already counted in a separate pile or systematically counting in one direction. They can be taught that one word goes with one item and to point to each item only once. Encouraging children to slow down can often reduce counting errors, as can increasing the size of spaces between items and using items that are not too small to hold and do not roll.

Counting objects laid out in a line is easier than objects scattered about. Once your child can count in a line, try practising counting objects laid out in different patterns, which require the child to keep track of counted items (e.g. in square, triangle, zig-zag or randomly arranged).

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Figure 9. Matching numeral cards

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Figure 10. Counting with a number line

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Figure 11. Matching cards to a number line

If your child has learned regular patterns for numbers (e.g. Numicon shapes) objects can sometimes be arranged into these patterns, so that the child can see, reinforce and use their knowledge about the relationship between pattern and number (see [Figure 5]).

Learning the number words in order

It is important that children learn the number words and how to count by rote. As well as counting during play, you can develop a daily routine to practise counting skills.

The following activities will support the first stages of learning to count and will be useful for supporting later understanding of ‘’how many’’.

• Matching numeral cards, learning to select them by name and name them ( [Figure 9]).
• Pointing to numbers on a number line to 10 and saying the number ( [Figure 10]).
• Matching numeral cards, to their position on a number line ( [Figure 11]).

Counting with a number line will help to establish the order of numbers and help children learn to say number words more clearly through practice.

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Figure 12. Matching numeral to a Numicon shape

Young children should first use the number line to learn the sequence of numbers to 10. Some children may also begin to practise hearing, distinguishing and saying numbers to 20 with a visual support, provided this does not confuse their learning of numbers to 10.

Learning about quantity

Visual cues for learning

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 children to learn this skill. Numicon activities can help, as the shapes indicate the quantity represented by each number.

Games to teach number and quantity with Numicon:

• Matching Numicon shapes (to 5) and then 10
• Matching numeral to a shape ( [Figure 12])

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###### Figure 13. Ordering shapes

• Selecting shapes by name
• Matching shape to a number line
• Ordering shapes ( [Figure 13])
• Matching shapes to appropriate quantities of pegs or other items

Numerals

Even before numerals can be identified reliably, their use may help children remember amounts, therefore use number cards or labels to place on containers or on the table as a prompt ( [Figure 14]).

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Figure 14. Numeral shown with set of 3 items

Matching quantities to numerals

Children can also be taught about quantities by using errorless learning methods, being offered only the correct amount of items to match to the numeral (or plate). For example, a child 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.

Children 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, children 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. For example, using the materials pictured in [Figure 14], the child is asked ‘’Can you give me 3 eggs?’’

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

Children 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 give some and leave the rest ( [Figure 15]).

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Figure 15. ‘Giving’ 3 items from a larger set

Games to practise counting part of a set and leaving some uncounted will help to teach this. Children 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 children may not receive enough modelling or practice in this type of activity. Usually early counting games require the child 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.

Children should be supported by a written numeral when asked for a set amount so that they do not forget the number they have been asked for. Having the number symbol in view for children will help them to remember and stop when their count matches the number requested.

‘’How many’’ covering and remembering games

When children 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 (eggs)’‘. Then cover the objects (or pictures of objects). Make it fun by saying’‘How many (eggs) am I hiding?’’ If they do not answer correctly, reveal the pictures or objects and say ‘’Look, there are two (eggs)’‘. When your child 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 child is helped to understand that counting tells us’‘how many’’ of something there is.

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 child 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 child 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 early number skills are often presented on worksheets or work books in the classroom ( [Figure 16]). Developing confidence with paper and pen activities can help even young children to work independently with a group. Children will be helped by becoming familiar with the ways work sheets present work and how they should respond to them. The responses typically required from children include circling numerals, number words or items, colouring them or drawing lines between them to associate or pair items or sets together.

! Examples of ‘home-made’ worksheets with use of colour ! Examples of ‘home-made’ worksheets with use of colour

Figure 16. Examples of ‘home-made’ worksheets with use of colour

! Using number stickers to record

Figure 17. Using number stickers to record

Children will also be able to practise learning how to write numerals and words through paper and pen activities. Many early number worksheets include these activities. Children will benefit from support to understand how to follow well designed, simple worksheets before applying their skills with more complicated worksheets. Graded Steps to Numeracy Books 1 to 10 TODO: references 8 are an example of worksheets designed for children with Down syndrome and other children who need a great deal of practice to master early numeracy skills and the writing of numbers.

Children can work with number stickers ( [Figure 17]) and other replacements for written numerals to demonstrate their understanding and skill with numbers. However, activities that teach them to write numbers and link numbers to quantity through paper and pencil work will supplement their understanding of number gained through activities with objects and apparatus. Developing writing and worksheet skills will help to prepare them for the style of work they will meet in later primary school years (8-11 years) and enable them to work independently.

Learning about ‘one more’ and ‘one less’

When children 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. Children with Down syndrome are likely to need more practice to understand these words 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 18]). Practice sums that use this language, interchanging ‘one more’ and ‘+ 1’. Children will be helped to use their skills by knowing the pattern of the number system forwards and backwards ( [Figure 19]).

! Word and symbol cards with apparatus and steps ! Word and symbol cards with apparatus and steps ! Word and symbol cards with apparatus and steps

Figure 18. Word and symbol cards with apparatus and steps

! Knowing the pattern of numbers to 20 backwards, forwards and in 2’s

Figure 19. Knowing the pattern of numbers to 20 backwards, forwards and in 2’s

Learning about bigger numbers

Children 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, so that they can discriminate them quite early on 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 20]). Using written words may help some children to discriminate and remember new words, for example, distinguishing ‘fifteen’ from ‘fifty’ (see Figure 29). 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.

For learning to say numbers and learning the order of numbers for use in counting, children 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 child. In the classroom a ‘spinner’ game or ‘roll the dice’ game can make choosing the beginning number more fun.

! A visual support for saying ‘13’ and remembering its place

Figure 20. A visual support for saying ‘13’ and remembering its place

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.

Addition

! A game to learn the stages of addition

Figure 21. A game to learn the stages of addition

Addition with objects and fingers

When children 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 adding ‘more’ means adding. Children can learn about adding and ‘add’ by joining groups of objects together to find out how many the new set makes.

Children 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 whole amount. They can also see how whole numbers can be broken into smaller parts.

Children can use various strategies for adding, usually beginning with the strategy of combining objects and counting them all, using fingers or other concrete materials.

A framework or game, with space for children to place numerals and objects or counters, can help them to learn the steps for addition ( [Figure 21]). Favourite characters can be included in games to make them fun to play ( [Figure 22]).

! Addition made interesting with a child’s favourite characters

Figure 22. Addition made interesting with a child’s favourite characters

! Symbol cards for a matching game

Figure 23. Symbol cards for a matching game

Discriminating and learning symbols

! Symbol cards with words on the back

Figure 24. Symbol cards with words on the back

The symbols for addition, subtraction, multiplication, division and equals, can be learned on cards through matching games ( [Figure 23]), with the words written on the reverse ( [Figure 24]). Symbols on worksheets can be highlighted or emphasised in other ways while children are learning them ( Figure 16).

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 children 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 children to ‘see’ how commutativity works.

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 children move up the number system and work with larger numbers. Learning doubles can be a fun activity for children 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. Children 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 children 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.

Counting-on

! Learning to count with a number line

Figure 25. Learning to count with a number line

Children can learn to count-on from one number, continuing their count sequence with the second group. Most children need to learn to carry on their counting without re-starting at 1, usually with the support of a number line ( [Figure 25]). Counting-on can be taught in a structured way and many children with Down syndrome aged 5-11 achieve this skill.

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.

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.

! 7 dots

Three sub-skills were then taught:

1. The children 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 her teaching cards.

! Skilled addition, using number facts, counting-on and fingers/objects to assist

Figure 26. Skilled addition, using number facts, counting-on and fingers/objects to assist

Teaching prompts included:

• saying the appropriate number names with the children, mouthing initial sounds and letting the children 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 emphasised the correct steps the child had taken).

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 26]).

Starting with the largest number

! Number square to help children see how tens work in the number system

! Number square to help children see how tens work in the number system

Figure 27. Number squares and tens cards to help children see how tens work in the number system

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.

Children 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, children 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 children can see when the units make a ten.

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. The same facts are being used repeatedly, but with bigger numbers.

Children 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. Like other children, 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 27]).

Mental arithmetic requires a good memory, especially when numbers are being split into tens and units for addition. Children 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 children with Down syndrome will need to record the steps to support their memory at each step in the calculation.

Children 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 children to check an answer they have found for a sum using a calculator and will practice place value.

Column addition is useful for finding out answers for 2 and 3 digit numbers. Many children with Down syndrome enjoy following procedures that they have learned to complete column addition. However, children 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. Children 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.

Learning about place value

! Word and number cards to aid discrimination and memory

Figure 28. Word and number cards to aid discrimination and memory

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, children need to be familiar with bigger numbers. After learning with numbers 1 to 10, children should learn to read and name numbers in order to 20, and then to see how tens fit onto a 100 square ( Figure 27). 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 children with Down syndrome.

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

Figure 29. Circling activity to assess discrimination of spoken ‘ten’ 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. Some teachers begin by teaching children the more logical way, in common with the rest of the number system, for example, ten-one, ten-two, ten-three. However, at some stage the usual way of reading and saying numbers will have to be learned.

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 children 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 28]). Discrimination of spoken ‘-ty’ words should be assessed ( [Figure 29]).

! Columns for learning place value

Figure 30. Columns for learning place value

Children 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 30]). Children can then practice matching the numerals in the number to their position or place.

Numicon activities can help children 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 children to find the shapes and to find the correct numeral symbol. Games to teach place value are provided with this equipment and embedded in the activities from the outset.

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

! Place value cards

Figure 31. Place value cards

Place value cards with a triangle or arrow next to the unit can help children to understand place value ( [Figure 31]). 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 placed on each other.

Games of exchange, using apparatus, help children to learn about hundreds tens and units and how these relate to each other. Some are bought games designed for this purpose and are available in most school resources, as all children need to learn about place value. 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 32]).

! A game of exchange: counting 12 in tens and units blocks

Figure 32. A game of exchange: counting 12 in tens and units blocks

An abacus can also be used to teach place value.

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

Subtraction

The idea of absence, removal, taking away, ‘one less’ and ‘nothing’ are less common in everyday life than ‘adding’ and children almost always find subtraction more difficult than addition.

It is likely that children 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 children with Down syndrome the same amount of practice that they had with addition if they are to understand subtraction. Children cannot see what has been taken away in subtraction in the same way as they can when things have been added. Children will not have practised counting backwards as often as they have practised counting and saying number words forwards, and counting forwards from one number to the next may be an easier way for them to ‘subtract’. They are likely to be more dependent upon written lines and squares for remembering where they are counting from and to and to remember which part of the series of steps is the ‘answer’ they need, i.e. the difference between the lower and higher number.

Practising counting forwards and backwards with number steps, number lines, and apparatus of various types will help children 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 children to add and subtract more easily, and be able to check their answer.

Strategies for subtracting include:

• Counting out the larger number using items or fingers, counting and then removing the smaller number and then counting what is left (a framework can help children to do this)
• 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
• Using a number fact and reversing it, for example, 4 + 4 = 8, so 8 - 4 = 4
• 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

! Subtraction framework

Figure 33. Subtraction framework

Subtraction with a number line also has a series of steps that can be learned through a framework.

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

Understanding how addition and subtraction are linked with each other will develop through practice. Children should also practice changing written sums around, breaking up numbers and joining them together (using apparatus), to help them understand adding and subtracting.

Just as children needed to know that ‘more’ means ‘add’ so they need to know that ‘less’ means ‘subtract’.

Multiplication

Children with Down syndrome can learn the same strategies as other children 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

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

Problems in words can be transferred on paper as a mixture of words and pictures or symbols to help the child understand the task. Children 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 34]).

When children know the steps involved in multiplication they can learn to apply their skills to functional activities, such as understanding and adding coin values ( [Figure 35]).

! A pattern translated into pictures and words

Figure 34. A pattern translated into pictures and words

! Demonstrating a multiplication strategy for life skills

Figure 35. Demonstrating a multiplication strategy for life skills

Multiplication using a calculator

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

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

First, children need to learn how to use a calculator. They can be helped to do this by following a list of the steps.

Children can practice creating their own sums, using a framework with blanks, for example, ‘There are (4) people and each would like (3) (apples)’. The numbers and words can be varied, for example:

There are ____ children. Each child wants ____ __________

Word cards can be varied, for example, apples, sweets, pennies, biscuits, balls, shells etc. and ready prepared on word cards, in the same way as the numerals on numeral cards.

The second part of the framework will be used to find out the answer, for example:

How many _____ do the children want altogether?

A framework can be laminated and used over and over again.

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

Pictures of adults, children and the items can also be placed on the card to make the activity more interesting and meaningful, for example:

Multiplication

Understand the problem:

You will be helping children find shells

1. There are 3 children. Each child wants 3 shells
2. How many shells do they want altogether?

Find the answer with a number sum: 3 x 3 = 9

Write the answer:

The children want 9 shells altogether

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 children will benefit from learning them.

Division

Children will benefit from sharing out items equally, or breaking up ‘whole’ numbers into ‘equal’ parts. The vocabulary for division should be used with children with emphasis, for example, ‘whole’, ‘parts’, ‘share’, ‘divide’ and ‘division’.

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

For the problem 15 ÷ 5, children 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

Children will learn about odd and even numbers and the significance of these for division. 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 developed an understanding of ‘odd’ and ‘even’ numbers.

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.

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. Children beginning to learn about division need to know that the bigger number is divided into groups of the smaller number.

Problem solving

Children can be helped to understand mathematics problems presented in words by drawing a mental representation of the problem (see [Figure 34]). Children may be helped by drawing or receiving help to draw the activity described in the problem. They can make lists of the important points and relate them to the picture. Just as flowcharts, lists, word webs, pictures and mental maps can help children with Down syndrome to understand text or remember complex information, so can these methods be used to help them understand problems, so that they can more easily decide how to solve them.

Most children with Down syndrome in the 5 - 11 age range will need help to understand problems and rearrange them so that they can more easily understand and solve them.

Measurement information

!

Measurements for length, weight and volume use the decimal system. Children will learn the terms 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 children who understand the decimal number system, as they can learn the word for each order of 10 through visual games and flash cards. Children 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

Children 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.

! A flip chart for school days

Figure 36. A flip chart for school days

Using a calendar, timetable or time line to begin to learn about time

Home-made calendars and timetables, that include written words, pictures or symbols for regularly occurring events, help children 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’.

! A calendar for learning time words

Figure 37. A calendar for learning time words

The complexity of the calendar will vary for each child, but you can begin with 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 36]). The calendar can be made more complex by having symbols or pictures and words for the separate activities of the days or evenings, when children are familiar with how to use the calendar.

Another type of calendar is illustrated in [Figure 37]. A pointer for ‘today’ can be moved along each day, and for older children, ‘yesterday’ and ‘tomorrow’ pointers can be added ( [Figure 37]). 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 child 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.

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

A simple 12 month wall calendar can be 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 children 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 child a real sense of the length of 2 weeks or 2 months or 2 days.

! A time diary

Figure 38. A time diary

Personal event or ‘time diary’ with photographs and sentences

To help children link themselves and their activities with time and time language, make books with the children 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 child will remember and then to apply the language of time (the day, the time, how long the event lasted, when the child may do this event again) in sentences written to go with the photographs ( [Figure 38]).

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

The time that children spend 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 children being able to make judgements about the ‘feel’ or passage of time. For most children, learning will continue into secondary school and for many years beyond this.

Within a whole-day time frame, children 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, 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. Children can ‘see’ how much time must pass before a certain event.

Children will be introduced to learning to tell the time from a clock face, with their peers when they are at infant school, around the age of 7. They may be helped to learn by having their own large cardboard 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, children 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).

Children 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 their skills become more advanced.

Similarly, at home children can learn the times in words and to recognise times on clock faces for their favourite television programme or other activities.

Children 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 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 children can make comparisons between the two.

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

Wearing a watch helps children to be more aware of the time, check times for events and practise telling the time. Children 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. Diving watches have an outer, moveable ring that has 10’s and often 5’s marked, so children with 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’ 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’’ to measuring time when cooking.

Children using a clock can have markers placed or stuck onto a clock face, so they can see when they are starting from and when the timed period will be complete, for example, for 10 minutes or 20 minutes for cooking cakes.

Timers can be easier to follow than clocks, for example sand timers, or more creative ‘timers’, that have been made and tested to last for a fixed time period. Sand timers are available for a variety of times and are helpful for showing time for completion of an activity. Timers can help children to finish activities, and to extend their turn taking skills.

Children can also use an analogue display timer, such as a cooker timer, 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.

! Infant activity with pennies

Figure 39. Infant activity with pennies

Money

Beginning to understand money

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

Children in the 5-11 age range should also handle real money, help with paying for goods and have their own money to take out with them. Children will learn to be responsible about money in school and outside of school by handling it. Many children, 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.

! Age-appropriate items and prices

Figure 40. Age-appropriate items and prices

Children first learn to recognise coins and notes by name. The numbers on coins are small so children 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. Children 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.

Children 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 ( [Figure 39]).

This will not prevent children 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 40]). In the infant years this matters less (age 5 to 8) but in the junior years, where children are aged between 8 and 12 years the appropriateness and the real cost of the items matter greatly. Children remember what they learn at school and to practice buying items for amounts that are not realistic is not helpful. Immersing children 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.

Learning how to add money, even to add 2p coins, will depend upon the children’s progress in learning about the number system and addition. If children do not understand cardinality and place value, trying to add coin values together will be very difficult. Even with these number skills children 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 41]).

! Teaching the value of 2p using coin cards

Figure 41. Teaching the value of 2p using coin cards

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 shapes to help children realise that the number on the coin tells us their value or worth, not their size, colour, shape or other features. When children 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. Children’s 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.

Exchange games

Children who know that one pound is 100p (or 100 pennies) can count money into ‘pounds’, making groups of 10p, 20p or 50p 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 child’s level of skill, is a suitable homework or weekend activity.

Pocket money

Children 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.

Giving the closest coin or note value in payment

! Early junior money worksheet

Figure 42. Early junior money worksheet

When children 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.

For accurately checking change, children need to be able to subtract and use the subtraction strategy of counting-on from the cost of the item, the lower number, to the amount they gave, the higher number. This is a sophisticated skill, especially in a real situation, but steps can be made towards this through learning experiences and number skill development in the primary 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
• 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 that notes are worth more than coins (as single items)
• Know that £5 note less than £10 note, £10 note less than £20
• Select coins and notes as needed
• Know about the existence of £50 note

Social learning

• Know the price of some everyday items that interest the individual
• Use vending machines
• Know about pocket money and when it is received
• 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.

Learning in the classroom

General principles for supporting numeracy

To learn in the classroom, activities will be simplified for children to varying degrees, and children will be following through the steps of a number programme. Adaptations will be made through teaching methods and resources to match children’s language, cognitive and social learning profiles.

In this section of the module for children aged 5-11 years, inclusion in the classroom, additional resources and for opportunities for extra practice are emphasised as particularly important for promoting learning and development for children with Down syndrome. Finally, recommendations for teaching number to children with Down syndrome are summarised.

Differentiation and a number programme

Across the maths curriculum, the authors expect children to make steady progress with learning new mathematical vocabulary, understanding number, diagrams, measurement and problem solving as they progress through primary school.

! Participating in class work on fractions

Figure 43. Participating in class work on fractions

Children will benefit from daily number work tailored to their individual learning needs in lessons, while other children are also completing their number work tailored to their individual needs. Much of the wider maths curriculum can be differentiated for children with Down syndrome so that they are included in activities and teaching designed for the whole class ( [Figure 43]). Differentiation is likely to be in language, complexity of the task, the style and amount presented, and in the size of the numbers in the task. Children can be supported by their peers and a support worker and use a wide range of classroom resources, some of them extra to the usual resources of the class. Children should also be encouraged to use their number and maths skills across the curriculum and in everyday life.

Inclusion in the maths curriculum will provide children with access to measurement, time and money from an early age, when typically developing children are learning these skills. As children progress through junior and secondary school they will need to continue to learn about time and money (as well as number) and relate these to daily living, age appropriately.

Teaching and supporting vocabulary, language and memory

Vocabulary

Reading, picture, gesture and symbol cues

Learning to read the word at the same stage that children are developing their 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, e.g. for prepositions (in, on, over, under, through, next to, behind, in front) or adjectives (narrow, wide, long, short, big, short, fat, thin, thick, full, empty) and so on. The word cards can be placed with objects, activities or games that teach the meanings of the words. Using the same word cards with different examples of the concept will help to generalise the children’s understanding.

For teaching comparatives, the card for ‘bigger than’ or ‘smaller than’ can be arranged with objects at either end, with the child ‘reading’ from left to right (e.g. The horse is ‘bigger than’ the dog). The objects can be swapped over to require a different (opposite) word card.

Reading number words will help the child to distinguish between words that sound similar when they hear them and that they may be discriminating between, for example ‘teens’ and ‘ty’ words. These words can be read and spoken aloud with different points of stress on the final syllables (see Vocabulary checklists at end of module).

Sentences and instructions

Sentences can be written onto strips of paper or card to help children remember what they are learning. Sentences can be broken into simpler sentences to help children follow the steps required more easily. Teaching materials and books, even for children who are delayed in learning maths and numeracy, do not necessarily allow for language delay so the language used may prevent the child from demonstrating their understanding and skills. Card and paper should be ready for support staff to write extra prompts or information that allow the child 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 children can tick or cross the steps off as they are completed.

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

Figure 44. Lift-the-flap charts to learn the order of numbers 1-5 and to count in 5’s

Memory training: Rehearsal and repetition

The rehearsal technique which is used in memory training activities can be used to teach children number facts ( [Figure 44]). The rehearsal technique is explained in full in the memory module in this series. The child learns the items in order by closing all the flaps then lifting the first one, naming the item, closing the flap and asking the child to name the item (now hidden). Next, the first and second items are uncovered, named, covered and the child is asked to recall the 2 items. The learning continues in this way until the child knows all 5 items. Consider applying this technique to digit cards, number lines and squares, tens tables, number words for rote counting, counting in two’s, five’s and ten’s, days of the week, months of the year, and regular activities in and out of school, with photographs, to teach time.

Maths practice

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

The following information comes from Sheila Hutchins, who describes her work supporting a 10 year old child (see [Figure 45]).

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 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.

! ‘Mixed maths’ for daily practice ! ‘Mixed maths’ for daily practice

Figure 45. ‘Mixed maths’ for daily practice

Resources

With good advance planning by the class teacher many children can be included in the class work, it is also important to ensure that additional resources and materials are available to teach the child with Down syndrome, as needed. These can be divided into resources for the individual which will be available for all classroom work, and a large bank of classroom resources.

Individual resources

These might include a ruler, numerals on stickers, card or in plastic, extra 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.

Whole school resources

Typically these include the visual and practical equipment for games and activities usually available in infant classrooms. In junior schools a reduced selection of these should still be available, as many children have delays in numeracy learning.

Resources in school might include:

• For number and measurement: number fans, number cards, bingo boards and cards, ball, magic screens, number lines to 20, number line games, flashcards, number square, number steps, spinner, rulers, metre rule, non-standard means of measuring, length flashcards, number and sign cards (+,-,=), question cards, prompt cards with written and numeral forms to 50, various objects, number strips, number stick, number cards to stick in books, objects for comparing, estimating, ordinal number cards and materials, felt pens, computer programs for number.
• For pattern and shape: sequence cards, pattern sequences, colouring sheets, shapes, logic blocks, partition cards, construction kits, cubes, multi-link, dice, dice cards, dominoes, paper dominoes, paper shapes.
• For money: coins (1p, 2p, 5p, 10p, 20p, 50p, £1), money cards, shop items, money worksheets, change worksheets, shopping cards.
• For time: Clock face, time flash cards (clocks and words), circular time sheet, chronological time sheet, date cards, question and answer cards, lists of months, numbered months, list of days, word cards, photography resources for making a time-diary.

In summary

The principles for teaching number to children with Down syndrome are the same as for all children - but taking account of language and working memory difficulties:

2. Introduce numbers and language for shape, size, colour, from 3 years of age, using matching, selecting and simple games
3. Teach the number sequence with the written numerals from the start - repetition is important
4. Play counting games to teach one-to-one correspondence
5. End with ‘’How many?’’ to encourage repetition of last count word to teach cardinality
6. Play counting games which end before the whole set has been counted - also to encourage understanding of cardinality
7. Play counting games that start at numbers other than one, once set has been counted, to prepare for addition and counting on
8. Being able to add is important for understanding place value (not just counting practice)
9. Use visual supports, digit cards, number lines, number squares and use materials which really do help the child to see the relationships between numbers (Numicon, Dienes)
10. There are no short cuts - a child must know the count sequence and then understand cardinality before they will be able to count-on. A child must be able to count-on to add and have much practise before they will understand commutativity and place value
11. Writing numbers helps child to understand place value in terms of how we write large numbers but addition helps child to understand 10 = 10 units, 5 = 5 units, 2 = 2 units, and then 12=10+2.
12. Until a child understands tens and units, they have no basis to cope with the decimal system for money or for weights and measures
13. However, children who do not manage to grasp the basic concepts will still be able to learn to use money, scales etc. by teachers adapting teaching targets in their teenage years
14. Small steps, much practice, visual supports for each step (mental maths will be very difficult given typical working memory spans)
15. Children struggling to understand the number system should still try all other areas of the maths curriculum e.g. simple fractions, geometry, plots, graphs

Vocabulary lists for numeracy

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 are included in Down Syndrome Education International’s vocabulary checklists as shown below.

• 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 children participate in activities, are presented in the vocabulary boxes.

Many more words are used in schools and a comprehensive vocabulary checklist for numeracy in UK 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.

Vocabulary list - Set one

Vocabulary list - Set two

Numbers and the number system

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. Computer software for number - some available in the Resources Catalogue of The Down Syndrome Educational Trust.
5. Wing, T. (2000). Serendipity, and a special need. Mathematics Teaching. 174, 27-30.
6. Geary, D.C. (1994). Children’s mathematical development: research and practical applications. Washington, USA: American Psychological Association.
7. Gelman, R. and Gallistel, R. (1978). The child’s understanding of number. Cambridge, MA, USA: Harvard University Press.
8. MacKinnon, C. (1998). Differentiated Learning Materials, 45 Carlibar Road, Barrhead, G78 1AD. Tel: 0141 580 1947.
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).

Acknowledgements

The authors would like to thank the children, parents, learning support assistants, teachers and schools they have worked with in schools across the UK, whose talents have contributed towards this module.

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