| Developing cognition which underpins work with numbers and maths
You have come to this site because you wish to help your child learn about numbers, however, I would like you to also think about the abilities required in order to acquire this learning. As a natural part of early childhood development children begin developing cognitive systems in other words ‘the infrastructure’ needed for this and other learning to take place. Another way to describe this infrastructure is our information processing systems, which are systems we all draw upon for everyday thinking.
Some of these terms you may have encountered during visits with your education psychologist or speech therapist. Children with Down syndrome do not develop these systems naturally therefore we need to maintain this focus so that we can stimulate and encourage this crucial area of development.
Dr Catherine Stern – mathematician, child psychologist and Montessorian, in corporated childhood developmental processes when developing her pedagogy and designing her supportive maths devices. Whilst working with the apparatus we are stimulating these processing systems. Every time your child places a block into a board or makes a pattern with cubes we are developing spatial thinking. When searching for a particular sized block from the randomly placed blocks on the table, he/she is also developing his/her ability to discriminate. The very visual impact of the games and by following simple instructions, children are stimulating both their visual and auditory processing ability. Children learn new vocabulary whilst working with the Counting Board (click for more information) as they elicit the meaning from their actions with this (CB) as well as reinforcement which the other equipment provides. “Which comes next?” “Which comes after this one?” “Which is the biggest or the smallest?” The new words come alive with meaning. We are also stimulating their memory ability, each game or activity in each session is using immediate memory, through constant repetition – their short-term memory, from the opportunity to visually input, from following simple directions or auditory input, and from handling or tactile input, these three routes transfer information into long-term memory storage for later recall and retrieval.
The Counting Board is a unique piece of apparatus. Its Gestalt (Click Gestalt for information) inspired design accommodates the stimulation of cognition as mentioned above, also the size of the blocks specifically encourage sensori-motor development. This is the period where infants learn from ‘concrete’ experiences through what they ‘see’ what they ‘hear’ what they ‘touch’ and from their sense of ‘smell’. Many children do not have sufficient experiences in the concrete, and are expected to move into abstract thinking and learning when they are not equipped to do so neither emotionally nor intellectually.
Young children with Down syndrome need a much longer period of concrete learning for these very reasons. This is also true of older children with Down syndrome where their intellectual ability and emotional levels are delayed. Further to this, there will be difficulties with manipulation or fine-motor ability. Often children with Down syndrome have chubby fingers so rods of 1cm size, are too small for them to manipulate, secondary to this is the need to internalise the value of each rod which has no clue to gain this information apart from their colour. Additionally this factor delays the number work they are being used for. It will cause frustration, as the child may not be able to position the rods correctly, resulting in negative behaviours to learning, such as refusing to cooperate. It can switch a child off maths, when this can and should be a fun motivational experience. A well known fact that motivation enhances real learning experiences!
 Another aspect of Stern is that apart from the exactness of the size of the blocks to encourage manipulation, they are graded unit blocks, for example the 3-block, is made up of 3 unit parts so children ‘see’ the value of the number implicit within. Also, the Counting Board device has a further crucial function, because it provides a structure that makes the learning as CERTAIN as possible! In the photo you will see how children acquire a breadth of linked knowledge through sight and touch. Take the example with 3, children not only see that it contains 3 unit parts (cardinal aspect) they can also see that it is in the third place (ordinal aspect). This gives children the opportunity to keep ‘filling’ the 3 with more defining aspects simultaneously, rather than the separate introductions of each aspect taught in isolation.
By filling the board with the various unit-block lengths an explicit picture emerges of where numbers ‘live’ in the series to 10. Small numbers are at the beginning, whilst bigger numbers are at the end - later as the child’s understanding of ‘left’ and ‘right’ increases, it will broaden this thinking. The visual image or pattern, also observed, ultimately demonstrates that each successive ‘number’ increases by one unit each time, or decreases in reverse order.
Later layers of teaching will link all knowledge gained with block work to abstract numeral s 1 – 10 bringing with it an emerging understanding or analysis of the basic numbers to 10. It becomes evident that a number such as 5 comes after 4 and is 1 unit bigger, or that 5 comes before 6 and is 1 unit smaller. Because the teaching structure of Stern is such that children do not have to work with abstract numerals initially, their early acquired number knowledge is focussed on learning about the properties, size, position, order and combinations of numbers to 10 all gained as a result of using concrete number representations in the form of blocks and patterns of cubes. Next comes the layer of learning where, imagine this, a numeral becomes the container with which to place all of the knowledge gained for each of the numerals.
Moving forward, children then have a wealth of understanding to draw upon when working completely with abstract presentations of addition and subtraction.
In contrast think about how children are regularly taught – they are introduced to numerals at home, reciting numbers to 10, some recognise these numerals, in or out of order, but get lost because they have no meaning. This continues at school where, through counting procedures, they parrot fashion a series of words that are meaningless. Children are exposed to abstract numerals with no clues to their meanings, count sets of counters or objects, first one set, then the second set, then a third count to find the total. So many things can impair learning, if a child hasn’t yet acquired 1:1 counting – one word for each object counted – totals are then incorrect, and the child has to repeat the count with no means of checking except from another uncertain counting procedure! Thus we create a situation whereby our children can spend years trying to crack the code…….
The examples above demonstrate the essence of Stern’s pedagogy and programme of teaching; the completeness of this breadth of knowledge comes from Stern’s ability to demonstrate the exactness and clarity of maths. Children are not enabled to think or reason mathematically from learning numbers by rote counting or in a piece-meal fashion where links and relationships cannot possibly be made. It is therefore so important to use this equipment as early as possible!
 The 10-Box - Interchanging your sessions with other apparatus such as the 10-Box will provide more reinforcement of this infrastructure alongside learning aspects of numbers. Spatial; discrimination; perceptual processing ability; size and position; fine-motor; memories; organisation of information;, sequencing, and so forth. Whilst in general children will begin work with number facts to 10 during reception and year 1, this can take many years for our children with DS to actually securely know their number facts to 10, from memory. Stern begins this learning process from the age of three in a very informal and nurturing manner.
Children begin discovering the ten combinations that total ten by fitting pairs of blocks into this device. The raft of information gathered from the games and activities includes, size and position of numbers, ordering numbers and concepts such as zero, as well as the comutativity principle - that it doesn’t matter in which order the two numbers are placed, the total does not change as in 1 and 9 or 9 and 1. Many children are not able to grasp this concept with general teaching and other resources. Stern takes them to this point in tiny progressive steps, layering their understanding until they are adding and subtracting numbers to 10.
Once the principles involved in working with  the numbers to 10 are known, you can give the child any box to learn the number combinations with sums less than ten. The black box (9) demonstrates that the 9-block fits by itself and needs nothing to make 9. 1 and 8, 2 and 7 and so on. With every even number box they learn the double number as well as further embedding the law of commutativity represented by like pairs of blocks in a different order. And of course the size of the blocks and cubes are encouraging sensory/fine motor activity continuously. This continuous manipulation also impacts on handwriting ability.
Speed of progress and achievement
When using the Stern programme it is usual to see some progress in one term. Imagine this for one moment - my daughter was ten years old before we began to see some real progress, given she went to school at four and a half, there were many wasted years, however, had I found Stern, when she was three she could have made significant progress. It is common for families to request advice from us to help their son or daughter who has become a teenager and is still not progressing with number and maths. Of course at this stage parents are usually anxious about their child gaining money and time skills before they leave school, both requiring basic knowledge of numbers.
You can see in the link below how an 8 year old with DS made extraordinary progress in just 6 months. Another example is with a teenager who moved to his secondary school with nothing in place in terms of maths ability. He too had been exposed to numicon but could not get on with it so spent the entire time in his primary school working at the P levels. It took this young man just 5 months to have securely learned his addition and subtraction facts to 10, had began to learn the value of coins to 10p and had moved on to work with teen numbers. He was twelve when he moved to the Stern programme, so we are looking at 7 wasted years of potential learning.
How these systems impact on other areas of learning
By stimulating these systems you will find that it will impact on all other areas of learning.
Children require spatial awareness when writing, to be able to leave spaces between letters in a word and between words in a sentence, to make correct margins at the beginning and end of a sentence, to throw or kick a ball, ride a bike, use a skate board. They would also require left/right directionality and sequencing ability to know in what order to place the letters correctly in a word and the words in a sentence as well as ordering numbers in the series, or to know the position of a 2-digit number commonly with teen numbers, where children write numbers in reverse order as with 31 for 13. Left right directionality is also necessary when learning to read.
In general children think in pictures so we need to help them develop their visualisation ability. Stern equipment provides unforgettable visual images which are stored in a child’s memory for them to draw upon even when they no longer need to use the apparatus. We all require good working memory, to remember shopping lists, or to follow instructions, to apply ourselves with immediate work, we need to be able to transfer information, procedures, and facts to long-term memory whatever the subject - remembering the timetable, daily activities in school and after school; which day is scouts, karate, swimming; how to set the dishwasher or put the washing machine on, how to follow a recipe - so we can recall the stored information for whatever reason in our daily lives. Therefore when looking to help your child learn his numbers, progress to mental calculation and other branches of mathematics - without consideration to the underlying infrastructure it will seriously impede your child’s development and as with many children with DS they may spend many years of unproductive learning.
For more information on the full teaching programme and equipment used please go to ‘our products’ tab.
Further reading:
http://mathsextra.wordpress.com/2010/05/24/from-theory-to-practice/
http://mathsextra.wordpress.com/2010/08/23/pupil-with-down-syndrome-achieves-with-maths/
If you would like help with specific information on teaching content, or to purchase the Stern programme to assist your child at home or at your child’s school, email enquiries@mathsextra.com or ring 0044(0)1747 861503
Families and schools in the USA or Canada can purchase the Stern programme from www.sternmath.com (based in New York). We would still be happy to provide helpful ideas and teaching support for your child with Down Syndrome and can be contacted via email: enquiries@mathsextra.com, via face book Vikki Horner - Maths Extra page, or the Maths Extra Blog.
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