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.
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.
Independent Educationalist, Retired Ofsted Inspector
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