Posts Tagged ‘Structural Arithmetic’

Cognitive Functions and Down Syndrome

December 17, 2010

“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


Learning whole numbers before units

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


The role of the teacher

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 http://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

Part 2 MEMORY GAMES

October 2, 2010

Memory game cards


This blog follows on from the blog dated 30/10/10 using Stern’s structured concrete apparatus to enable children to master the basic facts to 10 and less than 10.

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

How to play: place a set of cards in order 0 – 10 face up in front of the student(s).
Shuffle second set of cards and place face down. 1st player: picks a card reads it (7), parent/teacher says “7 and what make 10?” The student responds by saying the whole number sentence ‘7 and 3 make 10’ and places the card underneath the (3) in the ordered series. Continue until all cards have been placed correctly.

Number bonds to 10

To correct errors

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

Game 2:

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

After a few games the student may be ready to work with just the 10 card (maybe the sign, withdraw when ready) and a pile of cards face down. He says “10 subtract (turns over a card) (6) equals (4).” You can also recap by saying and pointing – “Let’s see what we have,” The student says 10 – 6 = 4.

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

NUMBER BONDS TO 10, LET’S GET THESE SORTED! (This is for Shaun and Michael!)

October 1, 2010

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

Let’s take a look at how a simple number box filled with number blocks can encourage this learning. The length of this box is the sum total of 8 and the blocks represent the numbers. By placing block pairs into the box children learn that there are 8 combinations that total 8. They can see that 8 comes after 7 and has one more unit. We can see the order and quantities of numbers 1 to 8, we can see the double or half, we could place 4 lots of 2s next to the 8 block to show its equivalence, from the block manipulation, processes are learnt, such as ‘bringing together’ two blocks to fit into the box we are describing and acting out addition, by taking away (out) one of the blocks of a pair we are describing and acting out subtraction. Both demonstrate the two operations as ‘doing’ and ‘undoing’, the relationship between these two operations. The box itself is integral to the learning because its role is to make the learning as CERTAIN as possible. The visual, therefore memorable aspect; the auditory aspect from following simple instructions; the hands-on or tactile experiences gained, are powerful routes to embedding information into long-term memory. Children are now enabled to recall facts efficiently from memory

Number bond to 8 in the 8-Box


Now we can move on to memory games with numeral cards and other resources

Memory games - materials

To add this step, we are helping children move away from concrete support. These games help with the transfer of knowledge learnt with the apparatus to semi-abstract learning with numerals. I have added this step as a means of continuing the learning without triggering ‘emotional barriers’ that get in the way of learning. When faced with formal work children pull away emotionally “Those numbers are too big mum.” “I can’t do that.” “I’ve never been able to do this.” “It’s hard.” It is my experience that even when you know a pupil has the skill in place, they can’t reach it. These barriers develop over time from not knowing what teachers are saying, from not knowing what they are supposed to be doing, from feeling stupid, being teased by their peers, from knowing they are failing. These children will often employ a range of avoidance tactics some of which can be disruptive.

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.

Short Term Memory and Working Memory

September 20, 2010

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

For instance, take the number six: the concept of six includes its ordinal value, its cardinal value and its relationships to other numbers, e.g. 2 x 3 = 6, 3 + 3 = 6, 3 x 2 = 6, 2 + 2 + 2 = 6, 5 + 1 = 6, 7 – 1 = 6, 12 divided by 2 = 6 etc., etc. From the ordinal value alone, or from the cardinal value alone or from one relationship alone, it is not possible to recognise or know the whole sixness of six, i.e. its totality.

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

As an example, take the completed 8-box: displayed is the ordinal value of eight, the cardinal value of eight, all the bonds that make eight, and some of its relationships, including its two halves. These are displayed visually and spatially simultaneously, as a whole display of concrete representations, and in a logical sequence. Together with the tactile features which contribute to the multi-sensory impact, and the range of materials which build up the bigger picture of the whole number system, these are what makes it so focused on learning through broad and flexible cognitive skills, rather than just on slow and frequently faulty counting techniques.

3. Practice

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

The enjoyment of exploring with Stern would be a significant positive effect, and success in achievement is a significant positive effect as well. Though lots of practice is needed, the Stern activities provide for experimentation and discovery, with plenty of variation in the exercises, thus minimising the boredom of frequent practice. Children gain enjoyment from recognising their own mastery and their own discoveries in using the apparatus. Take the example of an activity called ‘Climbing one’: a unit cube is placed on top of each block, 1 – 10, in the 10-box, and the ‘story’ of that bond is verbalised while looking at the display of the whole bond (done in logical sequence); then the unit cube is placed beneath each block in the stair, and the process repeated; finally the activity is then done with a two block (see photo) placed on top and underneath each block in the stair, as ‘Climbing two’. In my experience, at this stage, although there are no more ‘Climbings’ in the Stern programme, children start to generate ‘Climbing three’ and so on in a sequence, spontaneously themselves, calling the teacher’s attention to their discoveries and their competence with great delight.

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

Developed Cognitive Processing During a Single Term Using Stern Manipulatives – Part 2

September 11, 2010

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!”

Down Syndrome

September 4, 2010

Developing visio-spatial ability

Developing cognition to underpin work with numbers and maths

Children with SEN find mathematics one of the hardest subjects to grasp. In order to understand why this is so, we need to think about numbers. First, we are dealing with abstract concepts and an abstract number system. What are numbers? The dictionary defines a number as a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol. Numerals or symbols then are simply abstract configurations of lines and curves and yet with all that is known about the best means to teach children who struggle with maths, they are still being exposed to, and trying to make sense of, these symbols every day at school when what they need are visual, hands-on representations of numbers so that meanings and concepts can be truly grasped. From this concrete experience we can carefully lead them to abstract thought, first with memory games (numeral cards) until they are ready for formal presentations or abstract work.

Before we begin to explore the best ways to help children we need to give some thought to the abilities needed in order to acquire this learning.

Early Cognition

A natural part of early childhood is the development of cognitive systems. In other words, ‘the infrastructure’ they need for number work and other learning to take place. Another way to describe this infrastructure is our information processing systems (IPS). Systems we all draw on for everyday thinking and doing. I like to think about this in terms of a computer. We humans are the equivalent computer. When using the computer to create, for example, a power-point presentation the power-point software needs to be installed in the computer. This will enable you to create the most wonderful piece of work and retrieve it as often as required (stored in the computer memory). If the software is not there, how can you create your presentation? Now think about the software as our information processing systems; if there are deficits in these systems how are we able to draw on them to enable meaningful learning to take place?

Information processing systems

• Spatial ability
• Discrimination ability
• Sequencing ability including left/right directionality
• Visual and auditory perceptual processing
• Language ability – expressive and receptive
• Inter-sensory organisation
• Memory – working, short-term, long-term
• Sustaining attention

Some of these terms you may have encountered during visits with your education psychologist or speech therapist and may be a target within your child’s IEP and if so we can help to further develop this on a daily basis. Children with Down syndrome do not develop these systems naturally, so will have deficits with some or all of these processing systems. Therefore, maintaining a focus to develop these crucial abilities (IPS) is vital to a child’s learning.

IPS functions
Spatial awareness – simply put is an organized awareness of the objects in the space around us, and also an awareness of our body’s position in space. It requires that we integrate information from all of our senses. A classic example of developing spatial ability is seen where a toddler may be sitting with other children playing with toys. The toddler decides to go walkabout and simply tramples over toys and children because he/she is without awareness of the consequences. Later, we will see some development in awareness as this child now begins to negotiate around objects in h/her path. I recall an incident at my daughter’s school where, as she walked into the classroom, she tried to close the door without any awareness of the consequences or danger posed to the child following behind her. This naturally caused concern from staff from the potential danger of trapping small hands. Simply talking or explaining the consequences does not develop a child spatially.

Discrimination – is the ability to recognise or perceive differences. Babies tasting or putting things in their mouths is early discrimination.

Sequencing ability – is the action of following in order e.g. getting dressed. In terms of education, the ability to sequence is important when it comes to reading, writing and maths.

Inter-sensory organisation – is the brain organising information from all the senses.

Perceptual processing – is how we perceive information and process it through what we see, hear, smell, taste and touch. Children with perception difficulties will need time to process information. They need to experience the world through ALL OF THEIR SENSES – this is what is meant by the term multi-sensory learning.

Working memory – is memory needed for intermediate results, where information must be held during thinking.

Short-term memory – is what you can repeat immediately after perceiving it. Children with Down syndrome experience huge difficulty with memory, ask me I know, I would go over something that seemed like I had been doing it a thousand times and still it was not in place. Often teachers would say one minute it’s there, the next it’s gone! Think about this in terms of a ‘memory shelf’ where perceived information is placed. Some children have a ‘short shelf’ where information has the unfortunate habit of dropping off! This will help to understand why information that looks like it’s there one minute has gone the next… Passing information to long-term memory has to be meaningful and often needs repetition, but links between previously stored information and the incoming new information facilitates the process. Another factor can be overloading in the memory systems, we’ve all seen our children holding their heads!!

Long-term memory – is our ‘filing cabinet’ or our general store of remembered information. Once stored, it can be retrieved for immediate use.

Receptive language – is received information and can lag behind a child’s expressive language ability. Our children will have difficulties comprehending the subtle difference in meaning because same words have different meanings, and numerous words mean the same. There will also be difficulties when interpreting body language.

Expressive language – is the ability to verbalise ones understanding or needs.

Sustaining attention – is the ability to concentrate on some idea or activity, to stay on task.

The essential mechanism for ongoing thinking and immediate organisation is the working memory (Baddeley, 1997; Pickering, 2006). In their practical guide for teachers, Bristow et al (1999), describes structural apparatus (invented by Dr Catherine Stern) as a bridge into knowing and understanding number, and particularly recommends the Stern apparatus.

Encouraging the development of IPS

The stimulation of IPS is fully integrated in Stern’s pedagogy and support devices. Development begins alongside learning about numbers. Every time your child places a block into a board or makes a pattern with cubes, h/she is developing spatial thinking. When searching for a particular sized block from the randomly placed blocks on the table, he/she is also developing an 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. Language; children learn new vocabulary as they elicit the meaning from the teachers demonstrations and their actions with the equipment. “Which comes next?” “Which comes after this one?” “Bring me the biggest or the smallest block?” “Give me 5 and the block on bigger than 5.” 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 equipment or tactile input, from the interest and enjoyment these games provide are all routes helping to transfer information to long-term memory storage, for later recall and retrieval.

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’. Sensory input is especially important at the early stages of numeracy and is a vital aid to memory. Many children do not have sufficient concrete experiences, 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. Also because in general they are visual learners, they need a multi-sensory approach to provide an optimum learning environment. Stern provides children with a sensory-motor environment. Further to this, there will be difficulties with manipulation or fine-motor ability. Some programmes use Cuisenaire rods, or similar 1cm rods. These can be very fiddly and for some semi-abstract. Often children with Down syndrome have chubby fingers so rods of 1cm size are too small to manipulate, secondary to this is the need to internalise the value of each rod. A certain length or colour provides no visible clue therefore this factor delays the number work they are being used for. They can also 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 or ‘switching’ off maths which is a shame as it should be a fun experience!

Putting the theory into practice

The Counting Board is a unique piece of apparatus with many functions. In contrast to rods, Stern blocks are three-quarter inch and are size specific to encourage better manipulation and sensory development. They are also graded unit blocks, for example the 3-block, is made up of 3 unit parts enabling children to ‘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. Look at the 3, children not only see that it contains 3 unit parts (cardinal or quantity aspect) they can also see that it is in the third place (ordinal or position). 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 still, comes an emerging understanding or analysis of the basic numbers to 10, where 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. 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 connections, patterns and relationships cannot possibly be made. It is therefore important to use this equipment as early as possible.

Part two will provide some examples of pupils developing cognition and progress with numbers and maths …………..

Moving through Numicon to Stern

August 24, 2010

Hi Kathy

Thank you for your comments. In order to give you some individual support I would like to know a little bit about your child. What is the age and name of your child? What is the class/year group. Is it a mainstream or special school? Can you also tell me what your child already knows, you say there is some early number knowledge as well as knowing numbers to 100. This will help me identify gaps and show you how the Stern materials tackle this.

I am now going to respond as a parent so that I can talk about my personal experiences as a mother of a daughter with Down syndrome and the obstacles involved in my effort to help progress her maths skills, ultimately, following my long-term aim that she be able to use money! At the same time I will give some examples of the learning involved in securing sound basic knowledge and number facts up to 10 so as to give you a better idea of the programme and how it builds.

It was interesting to read that your experiences with the numicon materials paralleled ours and those of other parents who have contacted me with the same issues. We were one of the first families to use numicon! This was indeed the first resource that made a difference so much so that I began to spread the word to other parents and groups because I have always tried to pass on to other like minded families information/ideas/resources that have worked for us.

Our work with numicon
Internalising the patterns was not difficult because we built them gradually, nor was it difficult to understand the oddness or evenness of numbers. However, one of the main issues was that the steps were sometimes too big and so needed breaking down into smaller manageable ones. I also provided other activities to reinforce what was being learned with the shapes, all took time, and as you will also have experienced, repetition needs to be at the heart of learning. Over time it started to become apparent that there was a growing dependency with the shapes. During the sessions when the shapes were there it seemed that the number bonds to 10 were in place however I began to realise that the information had not internalised totally, and so when not present she could not retrieve all the number facts from memory. This is also the experience reported by other parents I have spoken to who were, like you, looking for some other manipulative in order to keep going.

We managed to keep going for about two years then my daughter lost interest in numicon and it became increasingly difficult to keep her motivated during our sessions which were mostly done at home because school didn’t want to follow this programme as I was doing. The reason I turned my attention to the Stern programme was simply because I knew numicon had based their ideas on the work of Dr Catherine Stern. Obviously my daughter was still my first priority. I had to keep searching for resources to help her. My search took me to New York, were I attended a weeks course on the use of Stern equipment which was very exciting. I saw some similarities but there were many more new ideas and pieces of equipment with a broad teaching base. I could see how I could build on what we had been doing. I came back with the whole set and got to work. The range of equipment definitely rekindled her motivation and keenness to learn.

Moving from one system to the next
Children work with two sets of number representations simultaneously – number blocks and cube patterns. The size of the blocks is a big plus. Because of Stern’s background as a child psychologist she designed size-specific blocks to encompass a child’s early sensori-and fine-motor development. These are on the ‘list’ in terms of our children’s developmental needs so we are doing something to stimulate this from the onset. I have seen lovely progress with strengthening fine-motor ability with older pupils which led to improved manipulation skills and hand-writing ability.

Rods versus blocks
Our problem with the rods used in the numicon programme, was that Charlotte could not grasp the values so I had to stop the teaching and work on learning rod values. This may not be the case for all the children? Would be interesting to hear how other children managed. By creating a sheet with a picture of each rod Charlotte had to match the rods to the picture and then later select until all rods had a name. This was time consuming and the bigger rods were never totally internalised. As well as this the rods are generally too tiny for our children’s little hands. Stern blocks are unit blocks of varying lengths, for example, you can see that the 3-block is made up of 3 unit cubes, and with the board you see that it is in the third place! So we are teaching both aspects together, the quantity of the number and its position.

Consolidation with the Counting Board

The first thing we did was quickly go through the activities with the Counting Board which is a wonderful piece of equipment to learn about the size and position of numbers to ten, their place in the series and ordering numbers to ten and reversed. It is very thorough. Charlotte very quickly learnt where each block lived in the board and was able to order them from the smallest to the biggest and in reverse order. This helped to reinforce early number knowledge and vocabulary. Attaching number names was also learnt quickly – at this time Charlotte was almost thirteen with more skills to bring to the maths session. For younger children it is a very nurturing and fun way to learn.

More ideas which are fun to get them thinking!

We played around with the board by putting a cube in the 10-groove, Charlotte was able to find the correct sized block to fill the gap (9). By the time we had completed this activity with the other blocks the visual pattern was immediate – that each block had a 1. Of course this game can be played when number names are known, thus it becomes an adding 1 or adding to 1 activity. Once the two blocks are in the gap say, whilst pointing to each block “So this is 9 and this is 1 and it makes 10,” With some children you can pause and let them say the name of the number themselves. You can also take the 10-block and get your child to place it on top then by the side of the two blocks to see that it measures the same or is equal to. Do the same with the other combinations. 8 and 1 makes 9, and so on. Reverse this by placing the single cube at the top of the board it changes the visual to 1 and 9 makes 10, 1 and 8 makes 9 and so forth. This is good preparation for later equation work.

Charlotte also liked to play the game where I asked her to fit two blocks into a certain groove (say 5) she would fit in maybe a 3, judge the space and find the appropriate block (2). Then I would ask her to move them both to the next groove (6 which would leave a space for one unit) initially I would point to that empty part and say “What about this bit?” Sometimes she simply added a cube making it 3, 2, 1, but other times she took out a block and replaced it with a block one bigger. Marvellous! This is a great way to help them to start reasoning.

Another activity - after Charlotte filled the board with the blocks, I would remove the 10 block and then take the 1 and place it at the front of the 10 empty groove. Next I would take the 2 and say look I am putting this behind it. (New word but the meaning is implicit). Next I would ask her to find the ‘next smallest’ and put it behind here (pointing) she would do this each time until we had built the stair. (Reinforcing the new word ‘behind’ every time). Also little fingers like to run up and down the stair, this is a simple way to check one-to-one correspondence, can the child name the block at the same time as the touch?

Broadening activity to introduce adding 1 to a number

I would take another unit cube and ask Charlotte to put it on the first step. “See what happens” She said “It’s the same as this one!” pointing to the two block. We continued up the stairs, (lovely and repetitious) providing a visual for each time you add 1 to a number it becomes the same as the next number! Notice I also changed my language from cardinal to ordinal, she was being exposed to both. “Put this on 1,” or 2, and so on. Put it on the first step,” the second step and so on. Children are enabled to learn maths vocabulary from the demonstrations and link it to their actions with the equipment. It is a very meaningful way of learning.

Stern Structured Patterns

The origins of the numicon shapes were that they were manufactured as a derivative of the Stern Pattern Boards, and in fact were still called Stern Plates when we began to use them. Thus the design replicated the same numerical structure as Stern’s Pattern Boards. Children move with ease from one to the other. The fact that one set of patterns are circular holes and the other is a set of inlays that hold cubes makes no difference. Once children have committed the patterns 1-10 to memory, it is the ordered structure that they see – the circle or square is irrelevant. For example with an array of 7 counters, sweets or tomatoes, can the child see a group of 4 and a group of 3 by drawing on their internalised pattern knowledge from either the Stern or numicon 4 or the 3 pattern? What about two 3’s and a 1? Can you see these sub-groups? This skill is important, and requires a lot of practice together with the knowledge of number facts to enable children to move away from counting to calculation.

In the next blog I will talk about how to help your child really internalise the addition and subtraction facts to 10 and less than 10, so our children can retrieve as recall from memory.

Please send me your comments or questions. Thank you Vikki

Pupil with Down Syndrome achieves with maths

August 23, 2010

8 year old girl with Down syndrome
In six months this pupil made some remarkable development in her understanding of maths and concepts.

The Stern maths programme
Structural Arithmetic takes a wonderfully practical and ‘hands on’ approach to Maths. This has helped to create visual understanding followed by conceptual understanding, and in the majority of topics this student has achieved such a solid grasp of what she is doing that she is regularly able to complete assessments without apparatus.

Working memory and long term memory strengthened
This scheme is so visual her working memory and her long term memory have both been stimulated and with regular reinforcement she is retaining the concepts and strategies over a greater period of time, and then building upon this knowledge.

Mental maths strategies encouraged
This pupil demonstrated emerging mental maths ability seen when ‘adding 9’successfully. She has retained number bonds to 10 and 20, and is using addition skills when moving through the decades with ease.

Skills learned through Stern are being applied to the traditional classroom maths text book giving this student access to mainstream maths, which is an obvious bonus. This young learner is thoroughly enjoying the scheme, reaping benefits from the positive impact that being successful and achieving can bring. Stern is a wonderful key to the Maths door way that is so often closed to many young learners with Down syndrome as well as pupil’s with other learning profiles who are struggling.

This is a wonderful example of how Structural Arithmetic helps children with Down syndrome develop maths ability with progress evident in a term. The above progress took place in 6 months. At 8 years of age, this pupil has another 8 years to continue achieving maths ability which is fantastic.

My problem is that many parents are not aware of the potential for learning that Structural Arithmetic can bring. Organisations that support families with children with Down syndrome seem to only promote the Numicon Approach thus making it difficult for parents to learn about other programmes that can help their child. To find out more about these resources see http://www.mathsextra.com or the Maths Extra page on facebook. I would welcome your comments or questions concerning your child’s levels of difficulty and how we can help.

Dorothy Latham – Independent Educationalist – former OFSTED Inspector

March 30, 2010

Dorothy Latham Says:

March 30, 2010 at 1:04 pm

I was very interested to read your fascinating summary of the work of Wertheimer, and his associates who influenced the creative thinking of Dr. Catherine Stern. As you mention, I used the Stern apparatus and its accompanying programme as the core scheme for mathematics teaching in both the schools of which I was headteacher, and found it excellent in concept formation and understanding of the number system in children’s minds. This comes through the natural discoveries made by pupils in using the apparatus in the activities, which they so often go on to creatively develop themselves. However, I need to point out that to achieve effective calculation at all levels of development, the load upon working memory must be reduced by memorisation of number bonds to an automated level through frequent practice. (The “counting on” and “counting back” method of performing operations impedes memory.)This is more easily done using the apparatus and the programme, because of the multi-sensory nature of the materials, the size of the materials (therefore having more impact), the enjoyable activities engaging children’s interest, the ease with which they can repeatedly record their findings, the opportunities for self-discovery, the facility for exhibiting relationships between numbers, the totality of the visual display “at a glance”. These all help to enhance the input into memory, apart from their value for the comprehensive grasp of the number system as a whole which children are able to acquire so effortlessly with Stern.

The essence of the Gestalt theory was the Gestalt concept of wholeness or totality, a stable and coherent whole or unified configuration, in which the parts by themselves have no independent meaning and cannot reveal the whole, but are dependent constituents of the whole. This concept has been applied to physical structures, physiological and psychological functions, and to symbolic structures, and is so obviously applicable to the structure of the number system, or indeed to any number system. Without knowing how the number five, say, relates to one, two, three or four, to seven for example, or even 35 or to 90 and so on, five has no meaning. Number is a system of relationships only fully meaningful within the totality of the whole system itself. The value of a number is dependent upon the way it relates to other constituents within the whole system.

It is this Gestalt aspect which has been so carefully and competently built in to the Stern materials and their programme of activities, through the displaying, revealing and discovering of relationships within the whole: within large sections of the whole to begin with and then building up to the realisation of the total whole.

Not only does children’s understanding become clear and comprehensive, but they enjoy their mathematical explorations and gain feelings of competence and success with using it.

I know Stern is regarded as very valuable in helping children with Special Educational Needs, but as someone who has used it in mainstream schools with the whole range of pupils, I wish that it could be seen as of core value to ALL children. Slow to start with, because of the need not to progress till the basic bonds are internalised in the memory, later it soon becomes an accelerator, hastening learning and ensuring quick and accurate calculation and supporting understanding of what numbers mean, essential in using and applying mathematics.

dorothy.latham@virgin.net
Dorothy Latham
1

Dr Catherine Stern and Gestalt Psychology

March 25, 2010

 In the previous post I stated that there was a link to Max Wertheimer and Gestalt theory and that little was known about his input with Sterns system except that it was indeed Max Wertheimer who named her approach Structural Arithmetic….. ….Well now we do!

Rather fortuitously I was recently invited to provide a Stern workshop at the National Association of Maths Advisors (NAMA) annual conference where the theme was: “MATHEMATICS BACK TO THE FUTURE – Learning from the past moving forward” This was a very appropriate platform to introduce Stern – past and present.

For those of you who are not aware, Stern’s Structural Arithmetic programme has been around for some 70 years, and 30 of those years the apparatus was used extensively in primary schools in the UK. It began to disappear from schools in the mid 80s, not because it was no longer effective in the classroom, but with the introduction of the National Curriculum which brought about a different emphasis to how we teach children mathematics. This change meant that good and effective concrete apparatus was relegated to the classroom cupboards to gather dust. Also, over time, the training in its use disappeared from Teacher Training Institutes and schools. I believe this was a classic case of ‘throwing the baby out with the bath water!”

 In preparing for this workshop, together with my continuing curiosity as to WHY Stern’s approach delivers such sound results, I began to investigate further – AND broaden and clarify my own understanding!

Sound credentials: Dr. Catherine Stern was, without a doubt, an extraordinary woman and achieved much recognition in her life time. Details of her life and works appeared in The Biographical Dictionary of Notable American Women by Richard D. Troxel. She was a mathematician, with a PhD in physics and mathematics,  an educator, Psychologist, Montessorian, as well as Consultant on Mathematics to the Carnegie Corporation. In essence, Catherine Stern devoted a life time to improving elementary maths teaching in the USA which had an impact on maths teaching in other countries especially England and Sweden.

Max Wertheimer – Founder of Gestalt Psychology -was one of the greatest 20th century scholars and theorists, who was the first to propose a Gestalt theory in psychology. This further evolved from the work of Wolfgang Kohler and Kurt Koffka. Many great scholars and theorists were associated with the New School of Social Research and were colleagues of Wertheimer: Karl Duncker (problem solving) George Katona (organising and memorising) Dr Catherine Stern (who explicitly applied ‘structural understanding’ to teaching), Abraham Luchins (problem solving) Albert Einstein (Theory of Relativity) who also showed a keen interest in the work of Stern. It was as a student of Wertheimer that Dr Stern amalgamated her search for more ACTIVE methods of teaching with Wertheimer’s Gestalt psychology. Her fundamental premise was that arithmetic should be taught so that the structural characteristics of the number system are understood and not just memorised.

From many years of study with children from early years, primary and secondary, Wertheimer remarked “Dr Catherine Stern has developed tools and methods for teaching arithmetic in which genuine discovery in tasks of a structural nature plays an essential role.”

This must have been a very exciting period in history where so many scholars gathered together in one place. Wertheimer’s ideas had an impact on the works of many great theorists and were of central importance in education and other fields such as social psychology, cognitive neuroscience, problem solving, art and visual neuroscience. However, it is said that the work of Katona and Stern probably came closest to his vision of Gestalt Theory of Education.

Throughout most of his career, the Psychology of Productive Thinking remained Wertheimer’s predominent focus where he called for greater co-operation and discovery learning in the classroom; that basic concepts should involve the nature and the role of structure in understanding.

  • Insight comes as an aspect of the discovery process
  • That the situation needs to be structured so as to make the necessary discovery as certain as possible!
  • The need to provide children with experiences in which a structure is evident, or by guiding them to the structure

Max Wertheimer’s greatest work was the Theory of Productive Thinking which was published after his death in 1943.

 Vikki Horner

My grateful thanks to Professor Brett King and Professor Michael Wertheimer – University of Colorado, Boulder USA

Recommended reading:
Historical extracts taken from:

MAX WERTHEIMER & GESTALT PSYCHOLOGY by D. Brett King and Michael Wertheimer

Published 2005 Transaction Publishers, New Brunswick, New Jersey www.transactionpub.com

NOTABLE AMERICAN WOMEN – Biographical Dictionary of by Richard D. Troxel.

Published 1980 Cambridge, Mass., & London: Harvard University Press


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