Support and Aspiration a New Approach to Special Educational Needs and Disability
When I began to read this paper I didn’t get very far before the following points resonated:
Children with special educational needs or a disability sometimes do not get the help they need to do well
Sometimes children with special educational needs or a disability do not do well at school because people do not expect them to
Parents have to ask separately for every bit of help they need
1.700.000 children with Special Educational Needs
Latest statistics state there are almost two million children in UK schools who have special educational needs; many will have difficulties with number and mathematics. Michael Gove, Minister for Education, yesterday suggested that pupils may have to stay on at school until 19 to retake GCSE’s in English and Mathematics in order to achieve a grade C. However, should we be trying to fix the problem at the top end? I don’t think so. Government should be focussing its efforts and finance at the bottom end, in Early Years because the problems begin here. We should be making sure we do all we can to help – ALL – children develop the pre-requisite skills that are central to learning BEFORE any formal education takes place.
Early childhood development
Simply put young children learn from concrete experiences in their world, from sight, hearing touch and smell, known as the sensory motor period. Cognitive functions are being stimulated and begin to develop. These functions, also known as information processing systems, are the infrastructure for yours, mine and every child’s ‘thinking’ and ‘doing’. For example, the way we perceive information both visual and auditory; spatial awareness; receptive and expressive language; discrimination; sequencing; inter-sensory organisation; sustaining attention; working and long-term memory. Children with some form of special need or disability will have deficits in one, some or all of these functions making meaningful learning almost impossible. Imagine how it would be for you if your memory was not working effectively?
Learning about numbers and mathematics
With this in mind let’s take a look at maths. The number system is based on abstract concepts. Numerals are symbols made up of configurations or lines and curves, and can be as meaningless to many children as it would be if I gave you a sheet of hieroglyphs and asked you to crack the code! Without the cognitive infrastructure children will not have intellectual nor emotional readiness to think in abstract terms. Think about the children with summer births, almost a year younger than their peers. If you think about the cognitive development taking place they would not have arrived at the same point of development and so would struggle to keep up. Parents report that their child’s teacher says don’t worry they will catch up in time but do they? Many teachers I speak to feel that the curriculum forces children to work with abstract concepts too soon and in their view this is where the problems begin. The maths curriculum is notorious for its fast pace, moving children on when clearly topics have not been grasped. Later would require some form of extra help. As pupils move up the school teachers receive children who do not have the required skills in place for continued development. The important building blocks required to withstand higher levels of maths are not being put in place. No wonder children become demoralised and disengage with the learning process? By the time they reach secondary education teachers have serious issues to deal with. Of course at this stage there are other factors to consider as to why students do not get high enough grades to warrant consideration such as is being proposed yesterday by the Education Minister.
We have to ask ourselves why the way maths is taught in schools for nearly two million children does not work and why we are continuing to waste huge numbers of years. I believe we are robbing children of many years of potential learning that they can never get back.
Personal Experience
I speak from personal experience, my second daughter was born with Down syndrome and wasted almost 8 years were little progress took place. When she was 6 her teacher suggested that she may have gone as far as she could because of her syndrome. I would not accept this and continued to search for a different route to helping her develop number skills. She was twelve when I found the Stern maths programme which has its roots in concrete learning through multi-sensory input. She handled blocks and created number patterns to build her basic knowledge and facts. We taught her for three years at home, with the last year in school. My daughter went on to pass GCSE at Entry level for maths. Whilst this was a cause for celebration I was also very angry that it was left to me to help her progress. I never gave up. I maintain that had Stern structured equipment, and this form of learning appropriate to her operating level, been in the classroom when she started school at four and a half she would have gained sufficient maths understanding and skills for adult life.
Maths Extra
Motivated by my personal experiences, my colleague and established a small company Maths Extra and have devoted the past seven years to helping other parents and teachers who work with a son or daughter or pupil(s) with SEN to provide information training and support with maths with some remarkable results. Even whilst this information is passed on to schools, we have found a mentality in some schools that leaves us speechless as seen with the following cases highlighted below.
Case 1 - A six year old pupil with Down syndrome – After personally providing weekly tuition for one term at home, reporting to the child’s school on progress, after a visit to the head to demonstrate how the equipment will help this pupil and others in the school, after loaning them a set of equipment which stayed in the school cupboard for five months, after an annual review where maths was a subject of concern, after follow up as to why they would not consider using this approach with this pupil I was told by the head teacher “I think we will stick with what we know.” Incredible!
Case 2 - Two years ago a parent of a 14 year old student with Down syndrome contacted us, following a recommendation from BIBIC to use the Stern programme in school because of the concrete and cognitive input embedded in the programme. School was sent copious information and yet took no action despite mums continued attempts to get them to implement this programme for her daughter. Her daughter has speech and language delay, cognitive deficits and fine motor delay and is operating within the P levels. In despair, mum came back to me in December 2010 deciding to make a start herself at home. They have already worked through the first levels of teaching and this student now enjoys maths and is progressing in just one single term, mum has reported noticeable increased maturity and increased fine-motor ability. The sad part of this story is that if school acted on the recommendations two years earlier, the student would have gained foundation number knowledge with basic facts to 10 in place. Currently she is still working within the P levels for maths. Two wasted years.
This student would have a statement and extra funding to support her provision thus it is not unrealistic for ‘mum’ to request they use this funding to purchase a resource that has been recommended by other professionals who are involved with her daughters development. The green paper is looking at giving parents more say in accessing the support they feel they need to help their child. I hope they do.
Positive progress
Contrast this case with a student of the same age and disability who moved to secondary school with no maths understanding and was operating at P levels. Five months after being put on the Stern programme, he had foundation understanding and bonds to 10 in place with increased cognitive development. Throughout his primary years his operational levels remained within P levels. Has the system robbed this pupil of his primary years of learning potential for maths? Yes it has!
Case 3 - This example involves a pupil with Autism in a mainstream school. After asking the school to look at the Stern programme for her son, the parent was told by his class teacher that it wouldn’t work for him, based on what? Did he take the time to found out about the Stern programme? No. The pupil is ten years old and has no understanding of numbers. Clearly what is being taught in school for this pupil is not working. Determined to help her son, ‘mum’ is just beginning to teach her child at home with the support of Maths Extra.
Dyslexia and Dyscalculia
Whilst dyslexia is commonly thought of as a language based difficulty between 50 and 60% of pupils with this learning profile will have surface issues with maths and can be due to cognitive deficits. Pupils who have dyscalculic tendencies would have more profound issues with learning about numbers and maths. These can be cognitive deficits, having little or no understanding of pattern, number structures and relationships. They would most certainly have what is often referred to as a ‘ones’ based number concept where their only ‘tool’ is counting. It is common for a child, for example, when asked the number that follows 62 to count from 1 all the way to 62 before saying 63. We have found that following the first teaching programme is a wonderful means to help these children progress. The use of specific equipment teaches pattern, structures and facts in a very visual and meaningful way, encourages the development of a number sense, understanding of number properties, and number facts up to and including 10. These will transfer to long-term memory due to the VAK element. We also see improved cognitive growth, which incidentally enables cross-curricula progress! This large group of pupils would really benefit from this multi-sensory means of learning as early as possible and certainly at the beginning of reception class.
How many years do some schools require before they acknowledge that what they are doing is not working. Do they honestly need 5, 6 or 8 years to come to this conclusion? These are wasted years that a pupil with some form of SEN can never get back. What has happened to the parent/school partnership? Government and schools are failing these children. Section 2 of the green paper sets out the need to give parents more say in the help they require to help there child and I look forward to seeing this implemented.
Stern Structural Arithmetic Programme
identifying numbers to 10
This programme was developed by psychologists and begins with the sensory motor period of learning for children aged three upward. It also incorporates natural stimulation of cognitive functions every time they are manipulating the equipment. Working with two sets of number representations in the form of graded number blocks to 10, a series of specifically designed number patterns, and mathematical devices, all aid the learning appropriate to the child’s stage of development. Children take in information through sight, hearing and touch in other words visual, auditory and kinaesthetic input VAK. Interestingly, Professor Usha Goswami – Director of the Department for Neuro-Science at Cambridge University stated two years ago, when maths was again a topic of discussion, that for optimal learning to take place yong children require continuous visual, auditory and kinaesthetic input.
Pupils who are still operating at this concrete level through delayed development no matter what age also need to work with this programme. There are no adult notions which require abstract thinking so soon, instead children learn through experimentation and trial and error, self correction, through curiosity and enjoyment gently building number knowledge and facts which transfer to long term memory storage. Quickly working with numbers as whole ideas before work begins with their unit parts encourages much quicker calculation ability from effective recall of facts. Early work develops a number sense (a crucial building block) cognitive development, number properties, size, position, order, number concepts, number relationships and bonds as well as encouraging a child’s reasoning ability. If ALL children, including children with a disability, in Early Years/reception had access to this means of learning outcomes at the end of key stage 1 would be much greater and the budgets greatly reduced. Whilst these two factors are important it is the child and their potential learning that deserves the first priority.
working with number patterns
finding bonds to 10
subtraction - problem solving
Extracts from the Stern teaching programme where all topics are supported by specifically designed maths devices to make numbers and number concepts visible. (Experimenting with Numbers Book 1 – Margaret B. Stern 2004)
More parents whose child is struggling to get help are resorting to taking action at home because they feel that school is not listening. In my view a parents roll is to reinforce what is happening in school and not to take the lead with teaching.
If you would like help and advice for home teaching, or to find out more information to help pupils with SEN in school please get in touch we are here to help.
enquiries@mathsextra.com
www.mathsextra.com
0044 (0)1747 861503
“Mummy, I HATE maths! Why can’t I do it?” “It’s not you honey – it’s the way you’re being taught……. “
I remember saying this to my daughter when she was nine. How many parents feel the same way? I know many of you do. The thing is, we wait patiently knowing our children have maths everyday, and yes, it is going to take them longer to get there because of their disability….. right? Then before we know it, years have gone by, and how much progress has been made?
This blog was prompted by another parent who contacted me because her child hasn’t made any progress for many years and is now 14. The bottom line is we have to ask ourselves why it is that we want our children, who have a learning disability, to have basic maths understanding? Isn’t it ultimately to transfer this knowledge to learning about time and using money? Don’t we wish to put these vital skills in place in order that our children can take more control of their lives?
The DSA information guide highlights the following cognitive difficulties:
• Delayed fine and gross motor-skills
• Auditory and visual impairment
• Speech and language impairment
• Short-term auditory memory
• Limited concentration span
• Difficulties with thinking and reasoning, and applying knowledge
• Sequencing difficulties
I would add to this list, lack of spatial ability, long-term memory difficulties and concentration – short span. Simply put, the above systems represent our information processing functions that underpin all thinking and learning. Thus children with Down syndrome will have deficits in one, some or all of the above and are a crucial factor in the lack of progress with maths (and other subjects).
Concrete experiences
Little children learn from their concrete experiences of their world, through all of their senses. It is logical therefore, that they should continue their development with concrete teaching until they develop the ability to enable them to grasp abstract thought. The very foundation of numbers and mathematics is built on abstract concepts, so we should make this information visible and tactile because children need to arrive at the stage of intellectual and emotional readiness before they can grasp abstract concepts.
Learning to read – whole words first
Years ago children were first taught the alphabet, then words of two letters, then words of three letters, and so on, until after much labour, they eventually learned to read. It was considered quite revolutionary when someone pointed out that ‘whole’ words could be learned without any previous knowledge of the alphabet. This was the route that DownsEd and other professionals advocated to help our children learn to read. This was the method I used to teach my own daughter to read when she was three. The alphabet did not come into our teaching and learning until much later.
working with whole numbers
We can see this same principle in the ideas and work of Dr Catherine Stern where she believed that it is easier and more in accordance with the child’s natural development, to learn ‘whole numbers’ before dealing with the units of which they are composed.
Stimulating cognitive processing functions
Structural Arithmetic
Cognitive processing is fundamental to all learning. I know I continue to allude to this in almost all of my posts with a maths focus, it’s because it is so important and because it is something that can get overlooked.
With Structural Arithmetic materials you can provide stimulation of every cognitive function on the above list, naturally every time you are playing the games within the Stern programme. This means every day in the maths lesson at school together with the drip-feed hoped for supporting this work at home! This stimulation is taking place alongside the building-up of your child’s number knowledge in this multi-sensory way. One of the consistent messages after a child begins to use the Stern programme, no matter what learning difference he or she may have, is that progress is seen with increased skill, number knowledge and cognitive processes in a single term.
Developing a sense of numbers
The first activities are specifically designed to develop a child’s sense of number from learning about number sizes and position – where numbers live in the series 1 to 10. As for cognitive stimulation, it would include scanning, judging and discrimination ability, visual and auditory perceptual processing, and memory.
Building the stair with blocks to 10, enables more practice with ordering the blocks – smallest to the biggest and in reverse order. This is providing more stimulation with sequencing, visual and auditory perceptual processing, memory.
The games and activities are motivational and engaging, thus concentration spans are increased.
Language practice
This is an important part of the learning, once the children understand how to perform a task, hitherto guided by the practitioner, h/she can take on this ‘teacher’ role demonstrated in the picture opposite. It is to encourage a child’s expressive language development. To be able to give instructions to other participants is a great way to practice emerging understanding and language.
This work is encouraging an understanding of both cardinal (quantity) and ordinal (order/position) aspects of number. Other activities will provide further opportunity to develop a sense of how two blocks of varying lengths combine to make 10, small blocks require bigger blocks and in reverse order, thus developing relational understanding.
As the names of the blocks are assigned, bond work begins as seen in the picture below.
number bonds
Each of the Stern maths devices is teaching and reinforcing number ideas, and is providing continuous stimulation of cognitive functions – scanning, judging sizes, position, sequencing, discrimination ability, working and long-term memory, visual and auditory perceptual processing. Language and concentration spans.
Working with, and internalising number patterns lays the foundation for addition and subtraction later on. The Pattern Boards will encourage cognitive growth. (Hand-eye co-ordination, one-to-one correspondence, left/right directionality). In the picture below we see children learning about the doubles.
Working with doubles
Throughout the programme children follow simple instructions that encourage receptive and expressive language. The size of the blocks, are specific to a child’s sensory motor development, continuous manipulation of the equipment increases fine-motor ability.
All round teaching and learning programmes
Stern programmes and teaching content can be found on www.mathsextra.com or contact 0044 (0)1747 861 503 for advice.
Obviously, more progress and development comes from regular use at home and at school. More and more schools are implementing the Stern programme. Contact us to see if your child’s school is one of them, or whether you would like us to contact them or provide any information. If you would like to chat about your child’s needs, I can be contacted on enquiries@mathsextra.com or on the above number.
Vikki Horner
It has always been the job of the educator to put abstract number relations into a concrete form which is adapted to the child’s interest and mental capacities. But while we adjust our teaching to fit the inner nature of the child, we must do so without damaging the inner nature of mathematics. Modern teaching attempts have so overdone the adjusting that the arithmetic itself is camouflaged and consequently poorly learned. (Dr Catherine Stern). Is this still true in today’s education climate? Is this because we are not following a child’s natural development? Why are children still struggling to learn number and arithmetic in schools today?
Stern Structural Arithmetic has been described as a bridge into knowing and understanding number (Bristow et al 1999). Stern’s method takes its start in activities natural to all young children, never artificially confronting them with adult conventions. These children discover number relations themselves as they must if relations are to serve them with full power and richness. They learn with delight; they continually show their teacher things they feel s/he must not miss. More than that, the number notions they acquire are not only adequate for the handling of ordinary experience, but are so sound logically, according to the best mathematical knowledge of our day that, as these children advance, they need never discard their early formulations as inadequate or deceptively simple, but can carry them forward as the building blocks of algebra. (Marguerite Lehr. Bryn Mawr College).
This blog gives examples of three pupils, one has severe autism, and two of the pupils have global discrimination delay. All of the pupils have specific processing deficits and have been seen to benefit significantly from being immersed within the Stern programme. Stimulating and encouraging a child’s information processing systems are a very important part of the pedagogy and maths devices within the Stern programme.
At a training course the maths coordinator of a school in Kent came to me and said “There’s no doubt about it Vikki, Stern is more than a maths tool.” This was a superb observation. His school began to use Stern with their Year 6 and 7 classes (within 6 months, they began to use Stern throughout the school). One year 7 pupil in particular held his interest.
specific number structure - Pattern Boards 1-10
The Counting Board - ordinal, cardinal aspects, and number relationships
“When the materials were first used with this pupil, I asked his art teacher to also monitor his progress in this subject in order to see if any changes were noted. Over the course of the term he was able to sort the blocks into their correct ‘channels’ and also master left/right sequencing and accurate copying up to the ’6′ pattern.”
An improvement in other curricula areas has been demonstrated through the development of this particular child’s cognitive learning systems.
“This pupil’s art work has shown considerable progress so much so that he won the end of term Senior Art Award for the most progress shown during the spring term. Progress is also being shown with his reading, for obvious reasons, and he is currently doing much to overcome ‘on/no’ reversals.”
“This I think is part of the benefit of the materials and the activities namely their cross curricula support because as this pupil’s global discriminatory skills increase these will feed back into supporting his mathematic abilities.”
“As for the year 6 class they have been using the materials at a ‘higher’ level to support 2 digit place value and staff have been very pleased with the children’s enjoyment of the equipment and progress it has brought.” Maths Co-ordinator – Special School Kent
Adding tens in the Dual Board
Sensory experience is essential in forming the imagery needed for internalisation of learning. While we are already familiar with VAK input (visual, auditory and kinaesthetic) as separate learning styles, Prof. Usha Goswami’s (Cambridge University), research has shown that creating a learning environment incorporating all three produces the optimum impact for learners.
Impact Learning – four routes to transfer information from working memory to long term memory
o Visual elaboration route
o Auditory route
o Kinaesthetic route
o Repetition route
Stern’s structural apparatus provides unforgettable imagery, therefore has visual impact. There is also sensory-motor impact through manipulation. The activities provide progressive repetition. There is opportunity for auditory and visual input through the teacher’s explanation; modelling; the child’s own commentary to peers; the incorporation of role play. This strengthens the child’s own developing mental strategies providing multi-sensory experience from the formation of imagery, building concepts and enhancing memory storage and recall.
A further reason for limited progress is the inability to work with the abstract nature of our number system thus many children are unable to access the maths curriculum when moved away from concrete experiences too soon.
The next two examples are a 9 year old boy who has severe global delay and also very little expressive language, and the other boy who is 8, is severely autistic.
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“Previously both boys would join in number rote counts and join in number rhymes with help but had no basic understanding of abstract mathematical concepts.” “Since the introduction of the Stern scheme there has been a marked improvement in both boys understanding. In the first case the boy is able to manipulate the blocks and place them vertically in their assigned place (At first he would place them horizontally and it took a good month to teach him to do it correctly). Now he recognizes the order they should be in and can match them to their twins.”
“With the autistic child who is a visual learner, he has moved very quickly through the first level (Stern) and has developed an understanding of ‘before’, ‘after’, ‘bigger’, ‘smaller’, ‘ordering’, ‘patterns’ and using blocks to make 10’s. He has now put number names to the blocks and has developed an idea of addition to 5. Modelling activities to learn the language, which for him is particularly difficult, and through his actions rather than the use of language, we can monitor his progress.”
“Both boys are developing a fundamental understanding of arithmetic concepts, which were lacking previously. It has also helped with their fine-motor and thinking skills.”
Special Needs teacher Primary School London
Stern has proved to be effective at any age, or stage of development. It has the ability to engage the learner’s interest, adding to long term memory storage effectiveness. It has the intrinsic capacity to display relationships between numbers clearly. The built-in, self-checking element of the apparatus automatically reinforces learning.
What is Structural Arithmetic?
Stern’s multi-sensory maths system was designed to develop a child’s emergent number sense by building-up number knowledge and number facts and by facilitating the understanding and use of the four number operations. Since the system is based on two tangible sets of number representations, the blocks and patterns promote a clear image of number in the concrete enabling pupil’s to discover for themselves all of the attributes on a physical level. When numerals are introduced they correspond to the blocks and patterns by embodying the intrinsic qualities and values of those numerals.
With our memory games we can move children away from the concrete support to practising and further embedding facts and concepts in long term memory, also to speed up recall of facts.
Stern is an inclusive system with proven success in early years and AEN. It aids the development of spatial thinking and reasoning, whether children are measuring blocks or working with patterns of cubes, they are applying spatial thinking to help them reason. Each experiment leaves a mental picture to turn around in their minds to explore new relationships.
Hand-eye coordination and one-to-one correspondence increases; unfamiliar maths vocabulary can be clarified by the practitioner through concrete demonstrations and further reinforced by the child’s actions with the apparatus; by following spoken directions children are able to develop receptive language; this is a means of assessing the child’s receptive language acquisition and auditory memory; a system of errorless learning where misconceptions in the child’s understanding will be seen immediately and acted upon through further demonstration and practice.
Catherine Stern had the vision to develop a maths system ahead of its time through tasks designed specifically to be SMART.
o Specific progressive steps which strengthens learning
o Measurable progress intrinsic to the teaching methodology
o Achievable with built in elements for success
- actively engages the child as a learner
- promotes the interest of the learner
- encourages expressive and receptive language
- enhances short and long-term memory
- develops opportunities for speaking and listening
o Realistic concrete small stepped programme incorporating scaffolded learning
o Time related – level of progress determined by individual
The ease with which practitioners are able to administer Stern’s system and speed of results, coupled with the immediate engagement and genuine enjoyment experienced by children is testament to the effectiveness of this maths programme.
Stanley builds a vertical staircase
Because Stanley liked building the stair we played the vertical stair game in the Counting Board. After placing all the blocks in the appropriate grooves, I took the 1-cube and placed it at the beginning of the 10-empty groove. Then I showed him the next smallest and he placed it behind the 1. After that I was able to ask for the next smallest each time which Stan placed behind the ever growing stair. In the middle of this I asked him to show me the biggest block where he pointed to the blue saying 10, next I asked which was the next biggest and he pointed to the black one (9). Although Stan was not able to use the number name he demonstrated understanding of ‘the next biggest’ That’s my boy! Because Stanley was still in his stride, I gave him a green cube asking him to place it on the first step. Of course he had to change it for the red cube! However he did place it on the step, Next I asked “What’s happened?” Although I was eagerly anticipating a response such as it’s the same as this one, he surprised me with “It’s the same as two!”
Counting Board ordering the number blocks
My continuous observations through this play session showed that Stanley is developing a sense of numbers, and that he is actively reasoning and revealing new learning through his own discoveries which is just what the Stern programme was designed to do. I look forward to my next visit!
I do hope more pre-schoolers are introduced to the world of numbers through these wonderful hands-on educational ‘toys’ who knows perhaps Father Christmas will deliver some down the chimney!!
Memory game cards
The ideas and games set out in part 2, assume that children are already demonstrating levels of accuracy with number bonds to 10 using the Stern blocks and activities from memory. What we need to do now is make the transition away from concrete support to working with numerals. Our memory games aim to encourage this practice from memory and speed up the process of recall. Each game provides a single focus, and can be repeated as often as required.
Practice with combinations to 10
Objective: reinforcing combinations – 1 and 9, 2 and 8, 3 and 7 and so forth
Materials: two sets of numeral cards 0 – 10
Games can be played with one student, paired or a group of students
Game 1
Practice cards - bonds to 10
Number bonds to 10
If a child picks say, the 7 card and says 7 and 2 make 10, the pupil can correct his own thinking using the 10-Box. We believe that when an error is made, it is more memorable for a child to figure out the correct partner for himself. When first moving to memory games have the 10-Box nearby, take out the 10-block and place it horizontally in front of the child. Ask him to get the blocks for his numbers (7 and 2). The child places these together alongside the 10 where any mistake will be evident.
Using blocks to correct error 7 and 2
Corrected number bond 8 and 2
Have the student repeat the correct number sentence successfully. Be aware of this combination and be sure to give further practice for this combination.
Tips: sometimes it’s fun for the teacher or parent to take a turn if working with only one child.
Extension game: each player simply points to the correct partner and keeps their cards. At the end of the game, they turn over each of their cards and say all combinations; 3, and 7 make 10, 4 and 6 make 10, 0 and 10 make 10 and so on.
NB: It is important to encourage the student to say the whole sentence and not simply the numeral.
Extension game: ‘Speedy is my brain’
‘WAS THAT HESITATION?’ (Named by one of my students)
Have 2 or 3 sets of cards to 10 shuffled and faced down. Each player selects a card and completes the number sentence. Everyone playing listens out for a pause and shouts hesitation! The teacher or another player says the number fact and the card is placed at the bottom of the pile. This game is full of motivation and fun especially when the teacher or parent joins in. When it is your turn, you of course have to pause so the students can catch you out! “Can someone help me out here?” … a player will tell you the fact you are ‘not sure about’ then it is placed at the bottom of the pile.
Group game: Individual practice. Have a stop watch or sand timer to hand. 1st player is given a set of shuffled cards where he says each combination as quickly as possible. Continue until each player has had a turn. Make a note of the time taken for each player. If there is time to play a second round the players are encouraged to beat their own scores!
Addition practice
Practice with addition and subtraction
Some students may work better initially with the familiar signs, so play a couple of games then see if they are okay at remembering which operation they are practicing.
Addition practice: Have two piles of cards, make sure that they are paired to make 10 before you place them face down. Place the 10 card after the equal sign where it remains constant. The first player turns over a card from the first pile, places it before the plus sign, say “1 + what equals 10?” After the player answers, he checks by turning over the card from the second pile.
Extension: Work with only one pile of cards, (2 sets shuffled). 1st player picks a card (7), names it and provides the addition to 10, “7 plus 3 equals 10.” At this stage the student is working from memory; the operation and the addition fact.
Subtraction using signs
Subtraction practice: first set up with minus and equal signs, placing the 10 on the left. The student turns a card (1) places it after the minus sign, reads the sentence, 10 minus 1 equals ….. then says the answer (9).
Subtraction from 10
Note: you may find that your student(s) may need more practice with reinforcing subtraction facts. Also vary maths vocabulary.
Finally, be creative and use other resources to maintain interest. Try tossing a die and say the number to make 10. Extend: toss the number die (6) then the die with a mix of + and – signs to see which operation to use. If it lands on the minus sign remind the student to begin with 10 can he complete the subtraction? You may have to prompt initially.
Practice with dice
This is a good way to assess whether your student is transferring these skills when using unfamiliar resources! Which is want we are trying to achieve ……..
Vikki Horner
Part 1
Why are number bonds important?
Why do children need to learn the basic facts to 10? – because, arithmetic skills are needed in other branches of mathematics. Why do schools allow children to move on without making sure that basic facts are secure? My daughter did not have these facts in place and yet she was given homework on fractions and decimals? Many parents have similar experiences with their child.
If your focus is on children with SEN then you would probably have a practical interest in your child’s number ability. As a mother of a daughter with Down syndrome, my passion was to help her understand maths to a level simply to enable her to learn to tell the time and use money. Anything else was a bonus!
To this end, your child needs to access the basic addition and subtraction facts to 10 and 20, from memory (efficient calculation) and be able to transfer these basic facts to other decades to 100. Simply put, this entails working with the number range 1 – 60 for time telling, and 1 – 100 for money (pennies in £1).
When we talk about our children learning their numbers, it is not just about being able to chant a string of numerals in order. Understanding numbers isn’t simply recognising or counting in differing quantities such as 1s, 2s, 5s or 10s and yet this is the ‘tool’ commonly used to gain arithmetical knowledge and skills. It is a long and laborious route.
Parents regularly report “My child knows his numbers but he doesn’t know what to do with them.” There lies the challenge! The ‘doing’ something with them implies an understanding of the properties and characteristics of each numeral and how to manipulate them.
The counting route can take years to master. A typical beginning comes with working with sets; say 3 counters and 5 counters. Children first count the quantity, 1 2 3 in the first set then count the quantity, 1 2 3 4 5 in the second set. There is of course a progressive process where children, in general, can be expected to arrive at ‘subitization,’ that is to see, at a glance, a quantity certainly up to five. This progression means a child will simply say the total (3) in the first set and count on the number of objects in the second set (4 5 6 7 8). My personal experiences with my daughter were that when asked how many altogether, she would arrive at the answer by counting 1 2 3 4 5 6 7 8.
Counting to figure out number facts
Let’s take a look at adding two numbers – if a child is asked the answer to 6 + 3 = ? and doesn’t know it, he figures it out by counting 1 2 3 4 5 6 7 8 9, or he may be able to hold the first digit in mind and say 6, then count on, 7,8,9. There is an assumption that encouraging children to count, count, count, will one day result in them stopping counting and say 9. However, for the counting child, six plus three does not equal 9, it makes nine by counting. Nothing in his mind makes this number fact unforgettable. Now what if he counts the total incorrectly as 10? He has no CERTAIN means of checking this answer except from a further UNCERTAIN counting procedure.
The shocking realisation is that we are seeing our teenagers not being able to move beyond this stage. I believe they are wasting valuable years of learning potential. Suddenly they are 8, 10, 12 or 18 years of age with still no knowledge of number bonds; a crucial tool to progress their basic +. –, facts to 10.
What is meant by the concept of number?
In order to understand the concept of a number, e.g. 8, we need to learn its ordinal value (8th place) its cardinal value (quantity – 8 unit parts), its position; 8 comes after 7 and is one unit bigger, 8 comes before 9 and is one unit smaller. Relationships with other numbers, 4+4=8, 4×2=8, 2×4=8, 2+2+2+2=8, 7+1=8, 3+5=8, 10-2=8, 16 divided by 2=8, 4 is half of 8 and so on. It is not possible to recognise the eightness of 8 in its totality, from its order alone (as with counting), nor from its quantity alone nor from knowing one relationship alone.
A quicker and more effective route – Learning through ‘measuring’
Dr Catherine Stern concluded that the ordinary counting approach does not lead pupils to see arithmetical relationships in a way that will permit relationships to be readily grasped. The piecemeal counting of single elements does not lead to the understanding of number relations. By substituting blocks of varying unit lengths, makes it possible for a child to measure instead of counting, where it is instantly apparent that a block that is 9 units long is the same length as the sum of a 3-block and a 6-block as with the earlier example, or a block that is 10 units long is the same length as the sum of a 6-block and a 4-block.
3 + 6 = 9
6 + 4 = 10
Number Boxes 1 to 10 – build a number concept of each numeral
Number boxes 1 - 9 Number bonds less than 10
Number bond to 8 in the 8-Box
Memory games - materials
Although children are only working with abstract numerals, they can cope with the games because they are fun and non-threatening. Memory games provide practice and reinforcement of number facts without the support of the blocks and facts are retrieved from long-term memory! Hurray………..
Note: What is shown above is only one strand focussing on number bonds, there are two other important pieces of equipment working on other areas, building up number knowledge as decribed above which can begin when your child is around three. You don’t have to wait until your child, over time, is failing in this regard, My advice- get started as soon as is possible. Imagine, literally working with number bonds and whole number ideas in a layered approach from around three years or even four years? I can, because I began to teach my daughter to read when she was almost three. At ten, she had a reading age of nine. I imagine often, how it would have been had I found Stern at that time and we had introduced these games into our daily Portage routines? All those years of struggling…… the up side of course is that, through my personal experiences trying to help my daughter has brought me to a point where I can now share our experiences and use my expertise to help you……
Vikki Horner
See Part 2 in the follow-on blog – practical ideas for memory games
Look out for a further blog, which shows how to use the knowledge of the facts to 10, when learning the bonds to 20.
After receiving the following comments to a recent blog post on Down Syndrome and Cognition, I decided to post this on the main site and not in the comments category because this in-depth theory and practice will be of interest to many blog readers. Vikki Horner
Short Term Memory and Working Memory
Dorothy Latham – Independent Educationalist
I am responding to the blog post on Down Syndrome and Cognitive skills, which is excellent. I especially like the way you have clearly distinguished and defined the Information Processing Systems and placed them in a useful bullet point list before explaining them. I think that everything you have delineated applies across the board to all children of all abilities.
I would like to comment on three areas and perhaps enlarge on these a little:-
1. Short Term Memory (STM) and Working Memory (WM)
STM and WM are almost synonymous, but not quite. Originally called Short Term Memory when this aspect of memory was first identified, it later became virtually replaced by the term Working Memory, due to a) the fact that people tend to use the term STM too loosely and not according to the experimental definition, and b) to the fact that researchers like Baddeley and others discovered further attributes linking STM to other cognitive and memory mechanisms, thus widening the concept itself. The model of WM now accepted is seen as encompassing two slave systems: a visuo-spatial sketch-pad (a temporary visual store) and a phonological loop (a temporary verbal store), both controlled by the central executive. (Baddeley and Hitch, 1974; Baddeley, 1986) While the central executive has a controlling and processing function, the slave systems aid WM as a whole to extend its range by rehearsal. However, there is experimental evidence (Baddeley, Gathercole and Papagno, 1998) to show that subvocal rehearsal to maintain information beyond the fast fade in seconds of STM doesn’t develop till around the age of seven. (Ages given when quoting research, are of course, average ages, since individuals may vary in development.) If such rehearsal is thus developmentally limited, how can young children manage successfully to do simple sums?
Bull and Espy (2006) (in Pickering, 2006) state ‘According to this developmental limitation, any verbally coded information, such as the addends of a sum, could not be rehearsed and, therefore, will be subject to rapid decay’. It is also thought that direct retrieval of arithmetic facts from Long Term Memory (LTM ) is improbable for young children, preventing accurate computation. Bull and Espy suggest that other functions from the central executive (CE) to do with attentional skills and the ability to switch these may be involved, as well as the involvement of the visuo-spatial scratch-pad (VSSP), since some young children do, in fact, solve such simple sums. The VSSP is likely to be a critical cognitive component in young children’s arithmetic, and one which has hitherto been ignored, say Bull and Espy. Other research has indicated that children progress from a stage of visuo-spatial use at around four years old, then through a transitional dual-use stage where verbal strategies as well as visuo-spatial ones begin to be used, before developing the more mature greater emphasis on verbal use, i.e. via the functions of the phonological loop (PL).
In 1995, Davis and Bamford studied the use of visual imagery by children of four to five when doing simple mental arithmetic. They varied the experimental conditions to include the use of concrete representations (small toys) or no concrete representations but only a mention of the toys, then added to some conditions the suggestion to the children to imagine a mental picture of the concrete representations used. Their findings showed more correct answers in the groups using the concrete representations, and where children were urged to use visual imagery after seeing the toys, their accuracy level was even higher. A number of other studies have also highlighted visuo-spatial skills as an important contributor to maths ability.
Mackenzie, Bull and Gray (2003) have shown that disruption to the VSSP, rather than to PL, reduces performance in arithmetic even for six year olds, indicating reliance on this slave system still at this stage. Thus young children appear to use a range of strategies within the functions of WM, and progress from physically counting concrete representations to visually recognising and imaging such representations, finally developing the ability to store and retrieve arithmetic facts directly from LTM. (Siegler, 1999)
For an image of a sum’s answer to become established in LTM, argue Geary, Brown and Samaranayake (1991), both the numbers given in the sum as well as the answer must be active in WM at the same time, enabling recognition and ultimately storage of the whole sum, i.e. the number bond involved (the Gestalt of the bond). Many studies show that children who do poorly in maths are still using immature procedural strategies and they take longer to solve calculations as well as making more errors; they find it difficult to move on from using counting to memory based problem solving functions. These children, when using slow and inefficient counting methods tend to lose information from WM, and imagery is not created in LTM. Using multi-sensory structural apparatus such as Stern, with its built-in visual and spatial aspects and the way all components of arithmetical bonds are simultaneously displayed in the completed activities, is obviously a most apposite way of helping young children not only to use their WM functions optimally to do simple arithmetic, but also actually to develop those functions.
2. The Concept of Gestalt
The original notion of the concept was derived from the German idea of ‘whole’, since the word Gestalt means a whole, a form or a shape. It is seen as something complete, coherent and stable. In its essential meaning it refers to ‘a totality that has, as a unified whole, properties that cannot be derived by summation from the parts and their relationships’ (English and English, 1958). On the other hand, the parts derive their properties from their membership of the whole, and the character of a mental concept of a whole is dependent on the way its constituents are combined in its organisation. The Gestalt concept has been used in psychology to identify the way that phenomena are organised and articulated. In terms of mathematical learning, the concept of a number as a whole depends upon the way its characteristic attributes are organised, although the separate attributes themselves alone do not indicate nor reflect the totality of the whole of the essence of that number.
6-Box ordinal and cardinal values and relationships
Thus there is much to be learned about a number before a complete knowledge and understanding of the full concept of that number is arrived at. Concepts of number bonds, and the formation of arithmetical processes, it would seem likely are formed in such a way. It is interesting that the argument put forward by Geary, Brown and Samaranayake (see above) that for the representation of an answer to a specific addition sum to become established in LTM the two addends must be present together with the answer in active form in the WM to enable storage of the bond to take place. Here, the complete bond is being conceptualised as a whole. Naturally, eventually this will contribute towards the establishment of the larger concepts of the numbers, the processes, and the number system itself.
The Stern apparatus was designed by Dr Catherine Stern with the stimulus of her knowledge of Gestalt theory, and its inbuilt features allow the visuo-spatial apprehension of ordinal, cardinal and relational aspects of the numbers to build the concepts of number bonds, of numbers themselves as wholes, and of processes. In the completed activities designed by Stern, the constituent parts of the components of numbers and number bonds are simultaneously displayed, thus aiding not only better understanding but more efficient storage and retrieval of learned facts presented in a logical sequence, which also aids understanding and learning since the sequence is itself part of a larger Gestalt.
8-Box bonds to 8
Finally, it is clear that ‘frequent revision will mean that concepts needed to provide meaning to new information will be readily available in LTM’ (Bull and Andrews, 2006). Thus retrieval will be more efficient and the processing load lower. Experience in the classroom endorses this. One application of a particular number bond or set of bonds is unlikely to result in effective or lasting storage, and multiple experiences of the same thing will be needed, varying from child to child, before stable storage and efficient retrieval are achieved. Children will actually comment ‘Don’t need them blocks any more, can do it in my head’ when they are ready to move on.
This aspect of learning recalls a very out of date behaviourist dictum I learned nearly 60 years ago, but which (although the Gestalt school of thinking was in fact in opposition to Behaviourism in terms of its philosophy) as a crude rule of thumb has stood me in good stead in structuring teaching and learning. (In those days it was called ‘The Laws of Learning’.) Today, they certainly wouldn’t, in this bald fashion, be regarded as a comprehensive or completely valid basis for understanding learning at all. Nevertheless, though learning involves so much more, these short tags draw attention to three key features which, within a larger context, still remain apposite for our practice. They are: i). Recency, ii). Frequency, iii). Effect. What this means is that you are more likely to remember something if it is recent, you are more likely to remember something if it has happened a number of times, and you are more likely to remember something if it has a significant effect. Significant effects may include happenings that were especially nice or especially horrible, or which have special sensory impact, or which have special meaning to you.
Climbing 2 - always next higher odd/even number
When I myself was a class teacher, teaching what were then called ‘top infants’ (equivalent now to Year 2) and using the apparatus as a core scheme, with daily practice for all children in the class, my higher achieving group were able by the end of the year to do successfully all four rules to hundreds, tens and units, albeit in division and multiplication with divisors and multipliers of only single figures. In fact, I remember introducing the concept of a thousand to them, and using the big thousand cube which was then available as part of the kit, subsequently going on to look at historical dates with them to subtract from the current year date to find out how many years ago certain events had happened. My middle group would certainly have been competent at, and at ease with, plus and minus at least in tens and units, and would have had a knowledge of numbers over a hundred and some idea of multiplying and dividing.
Dual Board - place value, composition, decomposition
The understanding of HTU and TU was taught using the Stern Dual Board, which exhibits so well the processes of composition and decomposition, the only logical methods of addition and subtraction. My lower achievers would have been working on internalising all the bonds of numbers between one and ten and the composition of teen numbers, with an introduction to twenty and over, using the twenty tray and the number track.
20-Tray teen numbers and teen facts
Number Track transferring basic facts
The important thing was to practise the basic bonds of all numbers between one and ten, both plus and minus (of which there are over 100) so that children understood the bonds and relationships between numbers, but also internalised them in consolidated and secure storage. We had the system that no child could go on to the next stage, at any stage, until perfect internalisation had taken place, and when I became a head teacher (of two schools, first an Infant school, and secondly an all-through primary school), we used Stern and I insisted on this rule. This meant that children were able to calculate quickly and correctly, without the errors which come with using a counting method rather than the method of stored, comparative imagery. Stern requires time to begin with, because of the need to keep practising until internalisation takes place, and this is different for all children, so individual monitoring is necessary; however, once the basic bonds are in place, progress becomes extremely rapid, due to the speed of calculation and the understanding achieved through the internalisation made possible by the use of Stern.
Although not often needed in mainstream KS2, if the foundation has been appropriately laid, Stern apparatus still does have some uses, and I have taught the squaring of numbers and square roots, the cubing of numbers and cube roots, to Year 6 using the mini-blocks (available once as part of Stern for older children). In one lesson, everyone got the understanding of what these terms meant so quickly and easily, because they could see and construct them for themselves.
Later when I became an Ofsted inspector, a role from which I have since retired, I had the opportunity to see children in a variety of classrooms under a variety of systems and using a variety of schemes and apparatus. Where children did well in maths, it was where they had achieved both understanding of the number system and the internalisation of the bonds, however this had been produced. Where children were struggling with maths, it was very obvious that they were relying on the business of counting on and counting back as a method of calculation, often still resorting to the use of fingers or number lines, instead of having practised their early bonds sufficiently to achieve effective storage of these in memory. This is not good enough and will always produce slow and faulty performance.
While the Numeracy Strategy had many good features (notably the mental maths for many children where bonds have already been internalised, and focus on different routes to aimed for answers) and helped many teachers and children through structuring and organising the teaching and learning of maths, to my mind it proliferated differing aspects of maths too early, interrupting the basic and vital process of sufficient and frequent practice of the bonds. It also split the knowledge of the bonds to ten between those up to five and those from five to ten, not a logical picture, and belying the knowledge of relationships between the numbers one to ten, which Stern does so well in beginning with the composition of ten. If teaching and learning could concentrate on achieving the internalisation of the basic bonds first, then all other aspects of maths would come quickly and easily in very little time at all.
I am passionately committed to the use of structural arithmetical apparatus for acquiring the understanding of number and the number system, and for using to practise forming the bonds until internalisation takes place. This is the key to success in maths. There are a number of different types of structural apparatus, but Stern and Numicon are the only ones which provide self corrective materials for developing the knowledge of number concepts and number bond concepts, described by Bristow et al (1999) as a bridge into knowledge. However, the Numicon system was actually originally based on the Stern apparatus, but without the chunkiness which gives Stern its strength in tactile aspects of its multi-sensory approach, and the visual and spatial impact of the larger units, as well as some pieces of equipment not copied in Numicon’s kit. Bristow et al state that the use of Stern’s material has been demonstrated as being effective with all children, having seen it in operation in some schools known to the authors. Having studied all the options, I feel sure that a grounding in maths using the Stern apparatus and following the Stern programme first, to achieve understanding of numbers, bonds and the number system with internalisation of bonds to automation point at all levels is the most effective way to develop children to become competent and successful mathematicians. As well as proving to embody the essential attributes for learning shown by the most modern research into developmental cognitive skills, it is attractive to children who enjoy handling it, and its structure promotes successful self-discovery.
A final comment
Though first designed with an almost historical idea at its roots, that of Gestalt, the Stern apparatus and its programme with its multi-sensory and discovery effects, and the way its simultaneous displays allow visuo-spatial apprehension of the whole and its attributes, is embracing effectively the most modern understanding of how children learn arithmetic. The latter picture was unknown to Dr. Stern at the time of her creation, but as a psychologist she was steeped in the careful observation of children as they learn, and forged her new learning system from her knowledge of what children could and couldn’t do, within the contextual influence of the idea that the whole is more than its parts, in a visionary way.
Dorothy Latham
Independent Educationalist, Retired Ofsted Inspector
References:
Baddeley, A. D., and Hitch, G. J. (1974) Working Memory, in Bower, G. A. (ed.), Recent advances in learning and motivation, (Vol.8, pp.47 – 90), New York, Academic Press
Badddeley, A. D., (1986) Working Memory, Oxford, Oxford University Press
Baddeley, A. D., Gathercole, S., and Papagno, C. (1998) The phonological loop as a language learning device, Psychological Review, 105, pp.158 – 173
Bristow, J., Cowley, P and Daines, B. (1999) Memory and Learning – A Practical Guide for Teachers, London, David Fulton.
Bull, R., and Espy, K. A. (2006) Working Memory, Executive Functioning, and Children’s Mathematics, in Pickering, S. J., (ed.) Working Memory and Education (2006) London, Academic Press
Davis, A., and Bamford, G. (1995) The effect of imagery on young children’s ability to solve simple arithmetic, Education Section Review, 19, pp.61 – 68
English, H.B., and English, A. (1958) A Comprehensive Dictionary of Psychological and Psychoanalytical Terms, London, Longman
Geary, D.C., Brown, S. C., and Samaranayake, V. A. (1991) Cognitive addition: a short longitudinal study of strategy choice and speed of processing differences in normal and mathematically disabled children, Developmental Psychology, 27, pp.175 – 192
Mackenzie, B., Bull, R., and Gray, C. (2003) The effects of phonological and visual-spatial interference on children’s arithmetic performance, Education and Child Psychology, 20, pp. 93 – 108
Pickering, S. J. (ed.) (2006) Working Memory and Education, London, Academic Press
Siegler, R. S. (1999) Strategic development, Trends in Cognitive Sciences, 3, pp.430 – 435
This is a very special post and is a story that will inspire any parent!
During the past nine months Angela has been very busy writing short stories. She has always been passionate about reading and enjoys acting out her favourite characters from the books she reads. What began as something to encourage her with writing, evolved into a much bigger project! Angela’s parents enjoyed her stories so much and realized that this recently developing ability with creative writing should be celebrated, so they decided to publish 20 of her best stories. What is more poignant to this story is that Angela has Down syndrome.
The Adventures of Edmond and Martha – by Angela Bettoni
Angela has always been a keen reader, which has obviously had an impact on her imagination and creativity, so to draw on these skills and create her own stories, is truly inspiring for someone so young.
Editing the series of stories has been kept to a minimum, so as to give a true idea of her writing. The stories themselves are amusing due to her great sense of humor! Each chapter is preceded by a scanned page of the original handwriting. The book is already on sale and can be obtained from Amazon priced £8.
http://www.amazon.co.uk/s/ref=nb_sb_noss?url=search-alias%3Daps&field-keywords=angela+bettoni&x=16&y=28
Background
Angela Bettoni is nine years old. She has had her fair share of medical problems, including heart surgery at 5 months to correct a VSD, ASD and PDA. She has also had pneumonia, bronchiolitis, innumerable chest infections and grommets. However, Angela has always been a very determined child, whose main love has been books from the time she was a year old!
Angela enjoys reading books by Enid Blyton, the “Naughtiest Girl” series; “Teddy Robinson”; “My naughty little sister” as well as picture books. Even though Angela is 9, her parents still make a point of reading to her at night and the Railway Children is currently taking its turn. Even when Angela was younger she had no problem remembering her stories. Because of this, or maybe to reinforce this, her parents came up with the idea for a game named ‘more books’ when travelling. They would quote a line from a book Angela had read then she would tell them which book it came from. What a great idea to reinforce thinking and memory ability!”
Because Angela also loves pretending, often in the Bettoni household one will have the pleasure of watching “Matilda”, “Ariel” or “Sophie” from Mamma Mia, performing, and because Angela pretty much runs these performances, her parents are also assigned roles and have to play along! When they hear “Lets’ pretend…” It’s show time…….!
In 2008 the Bettoni family moved to Rome, to a climate more conducive to Angela’s respiratory condition. They found a small British mainstream school where she joined a class (below her age group) of 15 children. Angela’s language work – reading, writing and comprehension – is at the level of her classmates.
Like many children with Down syndrome, Angela has struggled with mathematics. I first heard about Angela when her school contacted me to discuss using the Stern math’s programme to help develop Angela’s ability with maths. I was delighted to receive a report on her progress and achievement six months later.
Regular readers of my blogs will recall a post highlighting progress with a pupil with Down syndrome, this is Angela!
http://mathsextra.wordpress.com/2010/08/23/pupil-with-down-syndrome-achieves-with-maths/
More Special Schools Use Stern to Develop Cognition and Maths
Stern equipment is being used more and more in special schools, enabling pupils to make progress with their number skills where previously no progress was being made.
This pupil with global developmental delay, once introduced to the Stern programme, has shown dramatic results in some areas as reported by the maths coordinator:
“I have been focusing attention on the evident support that the apparatus can give to this child who has severe spatial and visual discrimination delay. Initially he was unable to place the blocks in the correct ‘channels’ in the Counting Board and was unable to copy the Pattern Boards and certainly not with left/right completion.” “When the materials were first used with this pupil, I asked his art teacher to also monitor his progress in this subject in order to see if any changes were noted. Over the course of the term he was able to sort the blocks into their correct ‘channels’ and also master left/right sequencing and accurate copying, up to the Stern ‘6’ pattern.”
An improvement in other curricula areas has been demonstrated through the development of this child’s cognitive learning systems. “His art has shown considerable progress so much so that he won the end of term Senior Art Award for the most progress shown during the term. Progress is also being shown with his reading, for obvious reasons, and he is currently doing much to overcome ‘on/no’ reversals.” “This I think is part of the benefit of the materials and the activities namely their cross curricula support because as this pupil’s global discriminatory skills increase these will feed back into supporting his mathematic abilities.”
For this pupil we are seeing increased spatial ability, from the progress made when working with the Counting Board and the creation of number patterns appropriately. We see his discrimination and judging size ability developing as he is able to correctly select blocks from the randomly placed blocks he has been using. His ability to sequence and demonstrate left/right orientation is also evident from the block ordering and from creating the Stern number patterns to 6. This emerging ability to orient from left to right has crossed over to being more able with writing. Fewer mistakes are being made when writing words commonly known as reversals where a child will write ‘on’ for ‘no’. The pupil’s visual perceptual processing has strengthened as indicated by his considerable progress with art work. This cognitive progress took place in one school term.
This school has also been using Stern with their year 6 class; using the materials at a ‘higher’ level to support 2 digit place value. They are very pleased with the children’s enjoyment of the equipment and progress it has brought. In the picture you can see why. It is easy to grasp the addition of two 2-digit numbers such as 30 + 20 = 50. The Dual Board with its separate compartments to hold ten single units on the right and another to accept ten whole tens on the left, mirrors abstract notation. By physically manipulating a number of 10s (blocks) seen here with three 10s (30), Stern is able to provide the learner with a clear image of quantity and structure aided by the apparatus and the manipulation. Thus adding a further two tens will, together, show the pupil he now has 5 lots of tens that total 50. the numerals below indicate the quantity of tens, the numerals at the top of the board indicate the totals. Learners have every clue to enable conceptualisation to take place.
The following two pupils attend a mainstream primary school in London. Both pupils have specific processing deficits and have significantly benefited from being immersed within the Stern programme. Here is a short account of what was happening during the first term.
Pupil with Severe Global Delay
“I started using Stern’s Structural Arithmetic with a 9 year old boy who has severe global delay and very little expressive language. Previously he would join in number rote counts and join in with number rhymes with help, however, he had no basic understanding of abstract mathematical concepts.” “Since the introduction of the scheme there has been a marked improvement in his understanding. This pupil is now able to manipulate the blocks and place them in their assigned places and can order the blocks 1 – 10. He can also match them to their twins.” (The first gentle step to understanding commutativity).
Pupil with Autism
“This pupil is a visual learner therefore Stern appeals to his strengths as a learner. He has moved very quickly through the first level (Stern) and has developed an understanding of ‘before’, ‘after’, ‘bigger’, ‘smaller’, ‘ordering’, ‘patterns’ and using blocks to make bonds to 10. He has now put number names to the blocks and has developed an idea of addition to 5. Modelling activities to learn the language, which for him is particularly difficult, also fits his profile. Through his actions rather than the use of language, we can monitor his progress. Both boys are developing a fundamental understanding of arithmetic concepts, which were previously lacking. It has also helped with their fine-motor and thinking skills.”
Stern has proved to be effective at any age, or stage of development. It has the ability to engage the learner’s interest, adding to long term memory storage effectiveness. The early work with the apparatus and simple activities provides a safe nurturing environment for children who need loads of support. The ‘puzzle stage’ working without number names gently immerses the child into a learning situation that does not crowd them with too many things to do at once. What we are teaching first, are number properties: about the size and position of numbers, as well as the combinations first with 10. Simple activities, where they measure a ‘gap’ with their eyes, then scan across a range of blocks, select an appropriate block to fill the gap. Of course if this is not the correct block, the child has immediate visual feedback from his action, as to why it does not fit, and leads him to a further action until he is successful. We never have to say ‘that’s wrong’, although I have watched a teacher (numerous times) in this situation say “Oh that’s too big, you need a smaller one” BEFORE the child has tried for himself. What did that teacher do? He took away a trial-and-error experience from the child and ultimately did that child’s thinking for him…. Making errors is a natural part of learning – we all do it! Making errors, makes the learning much more memorable!
Our small-step focus is on only one aspect, or part of a whole skill, at a time, get that in place, then we build and broaden by adding a further layer. Along-side this next aspect/objective we are enabling the child to use what has just been learned in the previous step. At each stage the child gets a real sense that he/she is achieving, where it is evident to both teachers and parents alike that confidence and self-esteem is growing. There is nothing more satisfying when your child says “Mum I know what to do!”
