---

Learning fundamental number skills

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

In the Numicon approach, a wide range of counting activities are advocated but the Numicon materials and activities have also been designed to support the development of mental imagery for whole numbers, which in turn will support mental arithmetic.

Drill and practice tends to be unfashionable in current teaching practice but there are good arguments for suggesting that children should practise the count word sequence until it is mastered to an automatic level, and similarly to learn multiplication tables and other useful addition skills (e.g. adding all combinations of 2 numbers, for 1 to 9, adding in 10's, 5's and 2's), so that they do not have to be consciously calculated when needed. Automatization of skills frees up space in working memory - the mental workspace used for calculations and problem solving.^[7]

A combination of a wide variety of counting and quantity experiences, the use of a visual image system to illustrate the ordinal nature of the system, place value and the relationships between numbers, and rote practice of number words, procedures for calculations and number facts, is probably the best approach.

Visual learning

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

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

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

Counting practice

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

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

Learning number words in order

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

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

Learning about quantity

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

Initially teenagers will learn to:

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

Matching quantities to numerals

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

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

'Giving' the whole set

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

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

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

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

'How many' covering and remembering games

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

Rearranging the same set, 'guessing' and counting again

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

Learning to write numerals, number words and to use worksheets

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

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

Learning about 'one more' and 'one less'

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

Learning about bigger numbers

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

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

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

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